4.9: The Cross Product
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Outcomes
- Compute the cross product and box product of vectors in \(\mathbb{R}^3\).
Recall that the dot product is one of two important products for vectors. The second type of product for vectors is called the cross product. It is important to note that the cross product is only defined in \(\mathbb{R}^{3}.\) First we discuss the geometric meaning and then a description in terms of coordinates is given, both of which are important. The geometric description is essential in order to understand the applications to physics and geometry while the coordinate description is necessary to compute the cross product.
Consider the following definition.
Definition \(\PageIndex{1}\): Right Hand System of Vectors
Three vectors, \(\vec{u},\vec{v},\vec{w}\) form a right hand system if when you extend the fingers of your right hand along the direction of vector \(\vec{u}\) and close them in the direction of \(\vec{v}\), the thumb points roughly in the direction of \(\vec{w}\).
For an example of a right handed system of vectors, see the following picture.
In this picture the vector \(\vec{w}\) points upwards from the plane determined by the other two vectors. Point the fingers of your right hand along \(\vec{u}\), and close them in the direction of \(\vec{v}\). Notice that if you extend the thumb on your right hand, it points in the direction of \(\vec{w}\).
You should consider how a right hand system would differ from a left hand system. Try using your left hand and you will see that the vector \(\vec{w}\) would need to point in the opposite direction.
Notice that the special vectors, \(\vec{i},\vec{j},\vec{k}\) will always form a right handed system. If you extend the fingers of your right hand along \(\vec{i}\) and close them in the direction \(\vec{j}\), the thumb points in the direction of \(\vec{k}\).
The following is the geometric description of the cross product. Recall that the dot product of two vectors results in a scalar. In contrast, the cross product results in a vector, as the product gives a direction as well as magnitude.
Definition \(\PageIndex{2}\): Geometric Definition of Cross Product
Let \(\vec{u}\) and \(\vec{v}\) be two vectors in \(\mathbb{R}^{3}.\) Then the cross product , written \(\vec{u}\times \vec{v}\), is defined by the following two rules.
- Its length is \[\| \vec{u}\times \vec{v}\| =\| \vec{u}\| \| \vec{v}\| \sin \theta, \nonumber \] where \(\theta\) is the included angle between \(\vec{u}\) and \(\vec{v}\).
- It is perpendicular to both \(\vec{u}\) and \(\vec{v}\), that is \[\left( \vec{u}\times \vec{v} \right) \cdot \vec{u}=0, \]\[\left( \vec{u}\times \vec{v} \right) \cdot \vec{v}=0, \nonumber\] and \[\vec{u},\vec{v},\vec{u}\times \vec{v} \nonumber\] form a right hand system.
The cross product of the special vectors \(\vec{i}, \vec{j}, \vec{k}\) is as follows. \[\begin{array}{cc} \vec{i}\times \vec{j}=\vec{k} & \vec{j}\times \vec{i}=-\vec{k} \\ \vec{k}\times \vec{i}=\vec{j} & \vec{i}\times \vec{k}=-\vec{j} \\ \vec{j}\times \vec{k}=\vec{i} & \vec{k}\times \vec{j}=-\vec{i} \end{array}\nonumber \] With this information, the following gives the coordinate description of the cross product.
Recall that the vector \(\vec{u}= \left[ \begin{array}{ccc} u_1 & u_2 & u_3 \end{array} \right]^T\) can be written in terms of \(\vec{i}, \vec{j}, \vec{k}\) as \(\vec{u}=u_{1}\vec{i}+u_{2}\vec{j}+u_{3}\vec{k}\).
Theorem \(\PageIndex{1}\): Coordinate Description of Cross Product
Let \(\vec{u}=u_{1}\vec{i}+u_{2}\vec{j}+u_{3}\vec{k}\) and \(\vec{v}=v_{1}\vec{i}+v_{2}\vec{j}+v_{3}\vec{k}\) be two vectors. Then
\[\begin{array}{c} \vec{u}\times \vec{v} =\left( u_{2}v_{3}-u_{3}v_{2}\right) \vec{i}-\left( u_{1}v_{3} - u_{3}v_{1}\right) \vec{j}+ \left( u_{1}v_{2}-u_{2}v_{1}\right) \vec{k} \label{crossprod1} \end{array}\]
Writing \(\vec{u} \times \vec{v}\) in the usual way, it is given by
\[\vec{u} \times \vec{v} = \left[ \begin{array}{r} u_{2}v_{3}-u_{3}v_{2} \\ -(u_{1}v_{3}-u_{3}v_{1}) \\ u_{1}v_{2}-u_{2}v_{1} \end{array} \right]\nonumber \]
We now prove this proposition.
- Proof
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From the above table and the properties of the cross product listed, \[\begin{aligned} \vec{u} \times \vec{v} &= \left( u_{1}\vec{i}+u_{2}\vec{j}+u_{3}\vec{k}\right) \times \left( v_{1}\vec{i}+v_{2}\vec{j}+v_{3}\vec{k}\right) \\ &= u_{1}v_{2}\vec{i}\times \vec{j}+u_{1}v_{3}\vec{i}\times \vec{k}+u_{2}v_{1}\vec{j}\times \vec{i}+ u_{2}v_{3}\vec{j}\times \vec{k}+ +u_{3}v_{1}\vec{k}\times \vec{i}+u_{3}v_{2}\vec{k}\times \vec{j} \\ &=u_{1}v_{2}\vec{k}-u_{1}v_{3}\vec{j}-u_{2}v_{1}\vec{k}+u_{2}v_{3} \vec{i}+u_{3}v_{1}\vec{j}-u_{3}v_{2}\vec{i} \\ &=\left( u_{2}v_{3}-u_{3}v_{2}\right) \vec{i}+\left( u_{3}v_{1}-u_{1}v_{3}\right) \vec{j}+\left( u_{1}v_{2}-u_{2}v_{1}\right) \vec{k} \end{aligned}\] \[\label{crossprod2}\]
There is another version of \(\eqref{crossprod1}\) which may be easier to remember. We can express the cross product as the determinant of a matrix, as follows.
\[\vec{u}\times \vec{v} = \left\vert \begin{array}{ccc} \vec{i} & \vec{j} & \vec{k} \\ u_{1} & u_{2} & u_{3} \\ v_{1} & v_{2} & v_{3} \end{array} \right\vert \label{crossprod3}\] Expanding the determinant along the top row yields \[\vec{i}\left( -1\right) ^{1+1}\left\vert \begin{array}{cc} u_{2} & u_{3} \\ v_{2} & v_{3} \end{array} \right\vert +\vec{j}\left( -1\right) ^{2+1}\left\vert \begin{array}{cc} u_{1} & u_{3} \\ v_{1} & v_{3} \end{array} \right\vert +\vec{k}\left( -1\right) ^{3+1}\left\vert \begin{array}{cc} u_{1} & u_{2} \\ v_{1} & v_{2} \end{array} \right\vert\nonumber \]\
\[=\vec{i}\left\vert \begin{array}{cc} u_{2} & u_{3} \\ v_{2} & v_{3} \end{array} \right\vert -\vec{j}\left\vert \begin{array}{cc} u_{1} & u_{3} \\ v_{1} & v_{3} \end{array} \right\vert +\vec{k}\left\vert \begin{array}{cc} u_{1} & u_{2} \\ v_{1} & v_{2} \end{array} \right\vert\nonumber \]
Expanding these determinants leads to \[\left( u_{2}v_{3}-u_{3}v_{2}\right) \vec{i}-\left( u_{1}v_{3}-u_{3}v_{1}\right) \vec{j}+\left( u_{1}v_{2}-u_{2}v_{1}\right) \vec{k} \nonumber \] which is the same as \(\eqref{crossprod2}\).
The cross product satisfies the following properties.
Proposition \(\PageIndex{1}\): Properties of the Cross Product
Let \(\vec{u}, \vec{v}, \vec{w}\) be vectors in \(\mathbb{R}^3\), and \(k\) a scalar. Then, the following properties of the cross product hold.
- \(\vec{u}\times \vec{v}= -\left( \vec{v}\times \vec{u}\right), \mbox{and} \; \vec{u}\times \vec{u}=\vec{0}\)
- \(\left( k \vec{u}\right)\times \vec{v}= k \left( \vec{u}\times \vec{v}\right) =\vec{u}\times \left( k \vec{v}\right)\)
- \(\vec{u}\times \left( \vec{v}+\vec{w}\right) =\vec{u}\times \vec{v}+\vec{u}\times \vec{w}\)
- \(\left( \vec{v}+\vec{w}\right) \times \vec{u}=\vec{v} \times \vec{u}+\vec{w}\times \vec{u}\)
- Proof
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Formula \(1.\) follows immediately from the definition. The vectors \(\vec{u}\times \vec{v}\) and \(\vec{v}\times \vec{u}\) have the same magnitude, \(\left\vert \vec{u}\right\vert \left\vert \vec{v}\right\vert \sin \theta ,\) and an application of the right hand rule shows they have opposite direction.
Formula \(2.\) is proven as follows. If \(k\) is a non-negative scalar, the direction of \(\left( k \vec{u}\right) \times \vec{v}\) is the same as the direction of \(\vec{u}\times \vec{v}, k \left( \vec{u}\times \vec{v}\right)\) and \(\vec{u}\times \left( k \vec{v}\right)\). The magnitude is \(k\) times the magnitude of \(\vec{u}\times \vec{v}\) which is the same as the magnitude of \(k \left( \vec{u}\times \vec{v}\right)\) and \(\vec{u}\times \left( k \vec{v}\right) .\) Using this yields equality in \(2\). In the case where \(k <0,\) everything works the same way except the vectors are all pointing in the opposite direction and you must multiply by \(\left\vert k \right\vert\) when comparing their magnitudes.
The distributive laws, \(3.\) and \(4.\), are much harder to establish. For now, it suffices to notice that if we know that \(3.\) is true, \(4.\) follows. Thus, assuming \(3.\), and using \(1.\), \[\begin{aligned} \left( \vec{v}+\vec{w}\right) \times \vec{u}& =-\vec{u}\times \left( \vec{v}+\vec{w}\right) \\ & =-\left( \vec{u}\times \vec{v}+\vec{u}\times \vec{w}\right) \\ & =\vec{v}\times \vec{u}+\vec{w}\times \vec{u}\end{aligned}\]
We will now look at an example of how to compute a cross product.
Example \(\PageIndex{1}\): Find a Cross Product
Find \(\vec{u} \times \vec{v}\) for the following vectors
\[\vec{u} = \left[ \begin{array}{r} 1 \\ -1 \\ 2 \end{array} \right], \vec{v} = \left[ \begin{array}{r} 3 \\ -2 \\ 1 \end{array} \right]\nonumber \]
Solution
Note that we can write \(\vec{u}, \vec{v}\) in terms of the special vectors \(\vec{i}, \vec{j}, \vec{k}\) as
\[\begin{array}{c} \vec{u} = \vec{i}-\vec{j}+2\vec{k} \\ \vec{v} = 3\vec{i}-2\vec{j}+\vec{k} \end{array}\nonumber \]
We will use the equation given by \(\eqref{crossprod3}\) to compute the cross product.
\[\vec{u} \times \vec{v} = \left\vert \begin{array}{rrr} \vec{i} & \vec{j} & \vec{k} \\ 1 & -1 & 2 \\ 3 & -2 & 1 \end{array} \right\vert =\left\vert \begin{array}{rr} -1 & 2 \\ -2 & 1 \end{array} \right\vert \vec{i}-\left\vert \begin{array}{rr} 1 & 2 \\ 3 & 1 \end{array} \right\vert \vec{j}+\left\vert \begin{array}{rr} 1 & -1 \\ 3 & -2 \end{array} \right\vert \vec{k}=3\vec{i}+5\vec{j}+\vec{k}\nonumber \]
We can write this result in the usual way, as \[\vec{u} \times \vec{v} = \left[ \begin{array}{r} 3 \\ 5 \\ 1 \end{array} \right]\nonumber \]
An important geometrical application of the cross product is as follows. The size of the cross product, \(\| \vec{u}\times \vec{v}\|\), is the area of the parallelogram determined by \(\vec{u}\) and \(\vec{v}\), as shown in the following picture.
We examine this concept in the following example.
Example \(\PageIndex{2}\): Area of a Parallelogram
Find the area of the parallelogram determined by the vectors \(\vec{u}\) and \(\vec{v}\) given by
\[\vec{u} = \left[ \begin{array}{r} 1 \\ -1 \\ 2 \end{array} \right], \vec{v} = \left[ \begin{array}{r} 3 \\ -2 \\ 1 \end{array} \right]\nonumber \]
Solution
Notice that these vectors are the same as the ones given in Example \(\PageIndex{1}\) . Recall from the geometric description of the cross product, that the area of the parallelogram is simply the magnitude of \(\vec{u} \times \vec{v}\). From Example \(\PageIndex{1}\) , \(\vec{u} \times \vec{v} = 3\vec{i}+5\vec{j}+\vec{k}\). We can also write this as
\[\vec{u} \times \vec{v} = \left[ \begin{array}{r} 3 \\ 5 \\ 1 \end{array} \right]\nonumber \]
Thus the area of the parallelogram is
\[\| \vec{u} \times \vec{v} \| = \sqrt{(3)(3) + (5)(5) + (1)(1)} = \sqrt{9+25+1}=\sqrt{35}\nonumber \]
We can also use this concept to find the area of a triangle. Consider the following example.
Example \(\PageIndex{3}\): Area of Triangle
Find the area of the triangle determined by the points \(\left(1, 2, 3 \right) , \left( 0,2,5\right), \left( 5,1, 2 \right)\)
Solution
This triangle is obtained by connecting the three points with lines. Picking \(\left( 1,2,3\right)\) as a starting point, there are two displacement vectors, \(\left[ \begin{array}{rrr} -1 & 0 & 2 \end{array} \right]^T\) and \(\left[ \begin{array}{rrr} 4 & -1 & -1 \end{array} \right]^T\). Notice that if we add either of these vectors to the position vector of the starting point, the result is the position vectors of the other two points. Now, the area of the triangle is half the area of the parallelogram determined by \(\left[ \begin{array}{rrr} -1 & 0 & 2 \end{array} \right]^T\) and \(\left[ \begin{array}{rrr} 4 & -1 & -1 \end{array} \right]^T.\) The required cross product is given by
\[\left[ \begin{array}{r} -1 \\ 0 \\ 2 \end{array} \right] \times \left[ \begin{array}{r} 4 \\ -1 \\ -1 \end{array} \right] = \left[ \begin{array}{rrr} 2 & 7 & 1 \end{array} \right]\nonumber \]
Taking the size of this vector gives the area of the parallelogram, given by
\[\sqrt{(2)(2) + (7)(7) + (1)(1)} = \sqrt{4+49+1} = \sqrt{54}\nonumber \] Hence the area of the triangle is \(\frac{1}{2}\sqrt{54}= \frac{3}{2}\sqrt{6}.\)
In general, if you have three points in \(\mathbb{R}^{3}, P,Q,R\), the area of the triangle is given by \[\frac{1}{2}\| \vec{PQ} \times \vec{PR} \|\nonumber \]
Recall that \(\vec{PQ}\) is the vector running from point \(P\) to point \(Q\).
In the next section, we explore another application of the cross product.
The Box Product
Recall that we can use the cross product to find the the area of a parallelogram. It follows that we can use the cross product together with the dot product to find the volume of a parallelepiped. We begin with a definition.
Definition \(\PageIndex{3}\): Parallelepiped
A parallelepiped determined by the three vectors, \(\vec{u},\vec{v}\), and \(\vec{w}\) consists of \[\left\{ r\vec{u}+s\vec{v}+t\vec{w}:r,s,t\in \left[ 0,1\right] \right\}\nonumber \]
That is, if you pick three numbers, \(r,s,\) and \(t\) each in \(\left[ 0,1\right]\) and form \(r\vec{u}+s\vec{v}+t\vec{w}\) then the collection of all such points makes up the parallelepiped determined by these three vectors.
The following is an example of a parallelepiped.
Notice that the base of the parallelepiped is the parallelogram determined by the vectors \(\vec{u}\) and \(\vec{v}\). Therefore, its area is equal to \(\| \vec{u}\times \vec{v} \|\). The height of the parallelepiped is \(\| \vec{w}\| \cos \theta\) where \(\theta\) is the angle shown in the picture between \(\vec{w}\) and \(\vec{u}\times \vec{v}\). The volume of this parallelepiped is the area of the base times the height which is just \[\| \vec{u}\times \vec{v}\| \| \vec{w}\| \cos \theta = \left( \vec{u}\times\vec{v}\right) \cdot \vec{w}\nonumber \] This expression is known as the box product and is sometimes written as \(\left[ \vec{u},\vec{v},\vec{w}\right] .\) You should consider what happens if you interchange the \(\vec{v}\) with the \(\vec{w}\) or the \(\vec{u}\) with the \(\vec{w}\). You can see geometrically from drawing pictures that this merely introduces a minus sign. In any case the box product of three vectors always equals either the volume of the parallelepiped determined by the three vectors or else \(-1\) times this volume.
Proposition \(\PageIndex{2}\): The Box Product
Let \(\vec{u}, \vec{v}, \vec{w}\) be three vectors in \(\mathbb{R}^n\) that define a parallelepiped. Then the volume of the parallelepiped is the absolute value of the box product, given by \[\left| \left(\vec{u}\times\vec{v}\right) \cdot \vec{w} \right|\nonumber \]
Consider an example of this concept.
Example \(\PageIndex{4}\): Volume of a Parallelepiped
Find the volume of the parallelepiped determined by the vectors
\[\vec{u} = \left[ \begin{array}{r} 1 \\ 2 \\ -5 \end{array} \right], \vec{v} = \left[ \begin{array}{r} 1 \\ 3 \\ -6 \end{array} \right], \vec{w} = \left[ \begin{array}{r} 3 \\ 2 \\ 3 \end{array} \right]\nonumber \]
Solution
According to the above discussion, pick any two of these vectors, take the cross product and then take the dot product of this with the third of these vectors. The result will be either the desired volume or \(-1\) times the desired volume. Therefore by taking the absolute value of the result, we obtain the volume.
We will take the cross product of \(\vec{u}\) and \(\vec{v}\). This is given by
\[\vec{u} \times \vec{v} = \left[ \begin{array}{r} 1 \\ 2 \\ -5 \end{array} \right] \times \left[ \begin{array}{r} 1 \\ 3 \\ -6 \end{array} \right]\nonumber \] \[=\left\vert \begin{array}{rrr} \vec{i} & \vec{j} & \vec{k} \\ 1 & 2 & -5 \\ 1 & 3 & -6 \end{array} \right\vert = 3\vec{i}+\vec{j}+\vec{k} = \left[ \begin{array}{r} 3 \\ 1 \\ 1 \end{array} \right]\nonumber \]
Now take the dot product of this vector with \(\vec{w}\) which yields \[\begin{aligned} (\vec{u} \times \vec{v}) \cdot \vec{w} &= \left[ \begin{array}{r} 3 \\ 1 \\ 1 \end{array} \right] \cdot \left[ \begin{array}{r} 3 \\ 2 \\ 3 \end{array} \right] \\ &=\left( 3\vec{i}+\vec{j}+\vec{k}\right) \cdot \left( 3\vec{i}+2\vec{j}+3\vec{k}\right) \\ &=9+2+3 \\ &=14\end{aligned}\]
This shows the volume of this parallelepiped is 14 cubic units.
There is a fundamental observation which comes directly from the geometric definitions of the cross product and the dot product.
Proposition \(\PageIndex{3}\): Order of the Product
Let \(\vec{u},\vec{v}\), and \(\vec{w}\) be vectors. Then \(\left( \vec{u}\times \vec{v}\right) \cdot \vec{w}=\vec{u}\cdot \left( \vec{v}\times \vec{w} \right) .\)
- Proof
-
This follows from observing that either \(\left( \vec{u}\times \vec{v}\right) \cdot \vec{w}\) and \(\vec{u}\cdot \left( \vec{v}\times \vec{w}\right)\) both give the volume of the parallelepiped or they both give \(-1\) times the volume.
Recall that we can express the cross product as the determinant of a particular matrix. It turns out that the same can be done for the box product. Suppose you have three vectors, \(\vec{u}=\left[ \begin{array}{rrr} a & b & c \end{array} \right]^T ,\vec{v}=\left[ \begin{array}{rrr} d & e & f \end{array} \right]^T ,\) and \(\vec{w}=\left[ \begin{array}{rrr} g & h & i \end{array} \right]^T .\) Then the box product \(\vec{u}\cdot \left(\vec{v}\times \vec{w}\right)\) is given by the following. \[\begin{aligned} \vec{u}\cdot \left(\vec{v}\times \vec{w}\right) &= \left[ \begin{array}{r} a \\ b \\ c \end{array} \right] \cdot \left| \begin{array}{rrr} \vec{i} & \vec{j} & \vec{k} \\ d & e & f \\ g & h & i \end{array} \right| \\ &=a\left| \begin{array}{rr} e & f \\ h & i \end{array} \right| -b\left| \begin{array}{rr} d & f \\ g & i \end{array} \right| +c\left| \begin{array}{rr} d & e \\ g & h \end{array} \right| \\ &= \det \left[ \begin{array}{rrr} a & b & c \\ d & e & f \\ g & h & i \end{array} \right] \end{aligned}\]
To take the box product, you can simply take the determinant of the matrix which results by letting the rows be the components of the given vectors in the order in which they occur in the box product.
This follows directly from the definition of the cross product given above and the way we expand determinants. Thus the volume of a parallelepiped determined by the vectors \(\vec{u},\vec{v},\vec{w}\) is just the absolute value of the above determinant.


11.8: Trait Theorists
Learning Objectives
- Discuss early trait theories of Cattell and Eysenck
- Discuss the Big Five factors and describe someone who is high and low on each of the five traits
Trait theorists believe personality can be understood via the approach that all people have certain traits , or characteristic ways of behaving. Do you tend to be sociable or shy? Passive or aggressive? Optimistic or pessimistic? Moody or even-tempered? Early trait theorists tried to describe all human personality traits. For example, one trait theorist, Gordon Allport (Allport & Odbert, 1936), found \(4,500\) words in the English language that could describe people. He organized these personality traits into three categories: cardinal traits, central traits, and secondary traits. A cardinal trait is one that dominates your entire personality, and hence your life—such as Ebenezer Scrooge’s greed and Mother Theresa’s altruism. Cardinal traits are not very common: Few people have personalities dominated by a single trait. Instead, our personalities typically are composed of multiple traits. Central traits are those that make up our personalities (such as loyal, kind, agreeable, friendly, sneaky, wild, and grouchy). Secondary traits are those that are not quite as obvious or as consistent as central traits. They are present under specific circumstances and include preferences and attitudes. For example, one person gets angry when people try to tickle him; another can only sleep on the left side of the bed; and yet another always orders her salad dressing on the side.
In an effort to make the list of traits more manageable, Raymond Cattell (1946, 1957) narrowed down the list to about \(171\) traits. However, saying that a trait is either present or absent does not accurately reflect a person’s uniqueness, because all of our personalities are actually made up of the same traits; we differ only in the degree to which each trait is expressed. Cattell (1957) identified 16 factors or dimensions of personality: warmth, reasoning, emotional stability, dominance, liveliness, rule-consciousness, social boldness, sensitivity, vigilance, abstractedness, privateness, apprehension, openness to change, self-reliance, perfectionism, and tension. See Table \(\PageIndex{1}\) below. He developed a personality assessment based on these \(16\) factors, called the 16PF. Instead of a trait being present or absent, each dimension is scored over a continuum, from high to low. For example, your level of warmth describes how warm, caring, and nice to others you are. If you score low on this index, you tend to be more distant and cold. A high score on this index signifies you are supportive and comforting.
| Factor | Low Score | High Score |
|---|---|---|
| Warmth | Reserved, detached | Outgoing, supportive |
| Intellect | Concrete thinker | Analytical |
| Emotional stability | Moody, irritable | Stable, calm |
| Aggressiveness | Docile, submissive | Controlling, dominant |
| Liveliness | Somber, prudent | Adventurous, spontaneous |
| Dutifulness | Unreliable | Conscientious |
| Social assertiveness | Shy, restrained | Uninhibited, bold |
| Sensitivity | Tough-minded | Sensitive, caring |
| Paranoia | Trusting | Suspicious |
| Abstractness | Conventional | Imaginative |
| Introversion | Open, straightforward | Private, shrewd |
| Anxiety | Confident | Apprehensive |
| Openmindedness | Closeminded, traditional | Curious, experimental |
| Independence | Outgoing, social | Self-sufficient |
| Perfectionism | Disorganized, casual | Organized, precise |
| Tension | Relaxed | Stressed |
Psychologists Hans and Sybil Eysenck were personality theorists (See figURE \(\PageIndex{1}\)) who focused on temperament , the inborn, genetically based personality differences that you studied earlier in the chapter. They believed personality is largely governed by biology. The Eysencks (Eysenck, 1990, 1992; Eysenck & Eysenck, 1963) viewed people as having two specific personality dimensions: extroversion/introversion and neuroticism/stability.
According to their theory, people high on the trait of extroversion are sociable and outgoing, and readily connect with others, whereas people high on the trait of introversion have a higher need to be alone, engage in solitary behaviors, and limit their interactions with others. In the neuroticism/stability dimension, people high on neuroticism tend to be anxious; they tend to have an overactive sympathetic nervous system and, even with low stress, their bodies and emotional state tend to go into a flight-or-fight reaction. In contrast, people high on stability tend to need more stimulation to activate their flight-or-fight reaction and are considered more emotionally stable. Based on these two dimensions, the Eysencks’ theory divides people into four quadrants. These quadrants are sometimes compared with the four temperaments described by the Greeks: melancholic, choleric, phlegmatic, and sanguine. See figure \(\PageIndex{2}\) below:
Later, the Eysencks added a third dimension: psychoticism versus superego control (Eysenck, Eysenck & Barrett, 1985). In this dimension, people who are high on psychoticism tend to be independent thinkers, cold, nonconformists, impulsive, antisocial, and hostile, whereas people who are high on superego control tend to have high impulse control—they are more altruistic, empathetic, cooperative, and conventional (Eysenck, Eysenck & Barrett, 1985).
While Cattell’s \(16\) factors may be too broad, the Eysenck’s two-factor system has been criticized for being too narrow. Another personality theory, called the Five Factor Model , effectively hits a middle ground, with its five factors referred to as the Big Five personality traits. It is the most popular theory in personality psychology today and the most accurate approximation of the basic trait dimensions (Funder, 2001). The five traits are openness to experience, conscientiousness, extroversion, agreeableness, and neuroticism (See figure \(\PageIndex{3}\)). A helpful way to remember the traits is by using the mnemonic OCEAN.
In the Five Factor Model, each person has each trait, but they occur along a spectrum. Openness to experience is characterized by imagination, feelings, actions, and ideas. People who score high on this trait tend to be curious and have a wide range of interests. Conscientiousness is characterized by competence, self-discipline, thoughtfulness, and achievement-striving (goal-directed behavior). People who score high on this trait are hardworking and dependable. Numerous studies have found a positive correlation between conscientiousness and academic success (Akomolafe, 2013; Chamorro-Premuzic & Furnham, 2008; Conrad & Patry, 2012; Noftle & Robins, 2007; Wagerman & Funder, 2007). Extroversion is characterized by sociability, assertiveness, excitement-seeking, and emotional expression. People who score high on this trait are usually described as outgoing and warm. Not surprisingly, people who score high on both extroversion and openness are more likely to participate in adventure and risky sports due to their curious and excitement-seeking nature (Tok, 2011). The fourth trait is agreeableness, which is the tendency to be pleasant, cooperative, trustworthy, and good-natured. People who score low on agreeableness tend to be described as rude and uncooperative, yet one recent study reported that men who scored low on this trait actually earned more money than men who were considered more agreeable (Judge, Livingston, & Hurst, 2012). The last of the Big Five traits is neuroticism, which is the tendency to experience negative emotions. People high on neuroticism tend to experience emotional instability and are characterized as angry, impulsive, and hostile. Watson and Clark (1984) found that people reporting high levels of neuroticism also tend to report feeling anxious and unhappy. In contrast, people who score low in neuroticism tend to be calm and even-tempered.
The Big Five personality factors each represent a range between two extremes. In reality, most of us tend to lie somewhere midway along the continuum of each factor, rather than at polar ends. It’s important to note that the Big Five traits are relatively stable over our lifespan, with some tendency for the traits to increase or decrease slightly. Researchers have found that conscientiousness increases through young adulthood into middle age, as we become better able to manage our personal relationships and careers (Donnellan & Lucas, 2008). Agreeableness also increases with age, peaking between 50 to 70 years (Terracciano, McCrae, Brant, & Costa, 2005). Neuroticism and extroversion tend to decline slightly with age (Donnellan & Lucas; Terracciano et al.). Additionally, The Big Five traits have been shown to exist across ethnicities, cultures, and ages, and may have substantial biological and genetic components (Jang, Livesley, & Vernon, 1996; Jang et al., 2006; McCrae & Costa, 1997; Schmitt et al., 2007).
Summary
Trait theorists attempt to explain our personality by identifying our stable characteristics and ways of behaving. They have identified important dimensions of personality. The Five Factor Model is the most widely accepted trait theory today. The five factors are openness, conscientiousness, extroversion, agreeableness, and neuroticism. These traits occur along a continuum.
Glossary
- Five Factor Model
- theory that personality is composed of five factors or traits, including openness, conscientiousness, extroversion, agreeableness, and neuroticism
- traits
- characteristic ways of behaving
Contributors and Attributions
-
Rose M. Spielman with many significant contributors. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the creative commons license and may not be reproduced without the prior and express written consent of Rice University. For questions regarding this license, please contact<EMAIL_ADDRESS>.Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>.


15.3: The Federal Reserve System
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2. How does the Federal Reserve manage the money supply?
Before the twentieth century, there was very little government regulation of the U.S. financial or monetary systems. In 1907, however, several large banks failed, creating a public panic that led worried depositors to withdraw their money from other banks. Soon many other banks had failed, and the U.S. banking system was near collapse. The panic of 1907 was so severe that Congress created the Federal Reserve System in 1913 to provide the nation with a more stable monetary and banking system.
The Federal Reserve System (commonly called the Fed ) is the central bank of the United States. The Fed’s primary mission is to oversee the nation’s monetary and credit system and to support the ongoing operation of America’s private-banking system. The Fed’s actions affect the interest rates banks charge businesses and consumers, help keep inflation under control, and ultimately stabilize the U.S. financial system. The Fed operates as an independent government entity. It derives its authority from Congress but its decisions do not have to be approved by the president, Congress, or any other government branch. However, Congress does periodically review the Fed’s activities, and the Fed must work within the economic framework established by the government.
The Fed consists of 12 district banks, each covering a specific geographic area. Exhibit 15.3 shows the 12 districts of the Federal Reserve. Each district has its own bank president who oversees operations within that district.
Originally, the Federal Reserve System was created to control the money supply, act as a borrowing source for banks, hold the deposits of member banks, and supervise banking practices. Its activities have since broadened, making it the most powerful financial institution in the United States. Today, four of the Federal Reserve System’s most important responsibilities are carrying out monetary policy, setting rules on credit, distributing currency, and making check clearing easier.
Carrying Out Monetary Policy
The most important function of the Federal Reserve System is carrying out monetary policy. The Federal Open Market Committee(FOMC) is the Fed policy-making body that meets eight times a year to make monetary policy decisions. It uses its power to change the money supply in order to control inflation and interest rates, increase employment, and influence economic activity. Three tools used by the Federal Reserve System in managing the money supply are open market operations, reserve requirements, and the discount rate. Table 15.3 summarizes the short-term effects of these tools on the economy.
Open market operations —the tool most frequently used by the Federal Reserve—involve the purchase or sale of U.S. government bonds. The U.S. Treasury issues bonds to obtain the extra money needed to run the government (if taxes and other revenues aren’t enough). In effect, Treasury bonds are long-term loans (five years or longer) made by businesses and individuals to the government. The Federal Reserve buys and sells these bonds for the Treasury. When the Federal Reserve buys bonds, it puts money into the economy. Banks have more money to lend, so they reduce interest rates, which generally stimulates economic activity. The opposite occurs when the Federal Reserve sells government bonds.
| Tool | Action | Effect on Money Supply | Effect on Interest Rates | Effect on Economic Activity |
|---|---|---|---|---|
| Open market operations | Buy government bonds | Increases | Lowers | Stimulates |
| Sell government bonds | Decreases | Raises | Slows Down | |
| Reserve requirements | Raise reserve requirements | Decreases | Raises | Slows Down |
| Lower reserve requirements | Increases | Lowers | Stimulates | |
| Discount rate | Raise discount rate | Decreases | Raises | Slows Down |
| Lower discount rate | Increases | Lowers | Stimulates |
Banks that are members of the Federal Reserve System must hold some of their deposits in cash in their vaults or in an account at a district bank. This reserve requirement ranges from 3 to 10 percent on different types of deposits. When the Federal Reserve raises the reserve requirement, banks must hold larger reserves and thus have less money to lend. As a result, interest rates rise, and economic activity slows down. Lowering the reserve requirement increases loanable funds, causes banks to lower interest rates, and stimulates the economy; however, the Federal Reserve seldom changes reserve requirements.
The Federal Reserve is called “the banker’s bank” because it lends money to banks that need it. The interest rate that the Federal Reserve charges its member banks is called the discount rate . When the discount rate is less than the cost of other sources of funds (such as certificates of deposit), commercial banks borrow from the Federal Reserve and then lend the funds at a higher rate to customers. The banks profit from the spread, or difference, between the rate they charge their customers and the rate paid to the Federal Reserve. Changes in the discount rate usually produce changes in the interest rate that banks charge their customers. The Federal Reserve raises the discount rate to slow down economic growth and lowers it to stimulate growth.
Setting Rules on Credit
Another activity of the Federal Reserve System is setting rules on credit. It controls the credit terms on some loans made by banks and other lending institutions. This power, called selective credit controls , includes consumer credit rules and margin requirements. Consumer credit rules establish the minimum down payments and maximum repayment periods for consumer loans. The Federal Reserve uses credit rules to slow or stimulate consumer credit purchases. Margin requirements specify the minimum amount of cash an investor must put up to buy securities or investment certificates issued by corporations or governments. The balance of the purchase cost can be financed through borrowing from a bank or brokerage firm. By lowering the margin requirement, the Federal Reserve stimulates securities trading. Raising the margin requirement slows trading.
Distributing Currency: Keeping the Cash Flowing
The Federal Reserve distributes the coins minted and the paper money printed by the U.S. Treasury to banks. Most paper money is in the form of Federal Reserve notes. Look at a dollar bill and you’ll see “Federal Reserve Note” at the top. The large letter seal on the left indicates which Federal Reserve Bank issued it. For example, bills bearing a D seal are issued by the Federal Reserve Bank of Cleveland, and those with an L seal are issued by the San Francisco district bank.
Making Check Clearing Easier
Another important activity of the Federal Reserve is processing and clearing checks between financial institutions. When a check is cashed at a financial institution other than the one holding the account on which the check is drawn, the Federal Reserve’s system lets that financial institution—even if distant from the institution holding the account on which the check is drawn—quickly convert the check into cash. Checks drawn on banks within the same Federal Reserve district are handled through the local Federal Reserve Bank using a series of bookkeeping entries to transfer funds between the financial institutions. The process is more complex for checks processed between different Federal Reserve districts.
The time between when the check is written and when the funds are deducted from the check writer’s account provides float. Float benefits the check writer by allowing it to retain the funds until the check clears—that is, when the funds are actually withdrawn from its accounts. Businesses open accounts at banks around the country that are known to have long check-clearing times. By “playing the float,” firms can keep their funds invested for several extra days, thus earning more money. To reduce this practice, in 1988 the Fed established maximum check-clearing times. However, as credit cards and other types of electronic payments have become more popular, the use of checks continues to decline. Responding to this decline, the Federal Reserve scaled back its check-processing facilities over the past decade. Current estimates suggest that the number of check payments has declined by two billion annually over the last couple of years and will continue to do so as more people use online banking and other electronic payment systems. 3
Managing the 2007–2009 Financial Crisis
Much has been written over the past decade about the global financial crisis that occurred between 2007 and 2009. Some suggest that without the Fed’s intervention, the U.S. economy would have slipped deeper into a financial depression that could have lasted years. Several missteps by banks, mortgage lenders, and other financial institutions, which included approving consumers for home mortgages they could not afford and then packaging those mortgages into high-risk financial products sold to investors, put the U.S. economy into serious financial trouble. 4
In the early 2000s, the housing industry was booming. Mortgage lenders were signing up consumers for mortgages that “on paper” they could afford. In many instances, lenders told consumers that based on their credit rating and other financial data, they could easily take the next step and buy a bigger house or maybe a vacation home because of the availability of mortgage money and low interest rates. When the U.S. housing bubble burst in late 2007, the value of real estate plummeted, and many consumers struggled to pay mortgages on houses no longer worth the value they borrowed to buy the properties, leaving their real estate investments “underwater.” Millions of consumers simply walked away from their houses, letting them go into foreclosure while filing personal bankruptcy. At the same time, the overall economy was going into a recession, and millions of people lost their jobs as companies tightened their belts to try to survive the financial upheaval affecting the United States as well as other countries across the globe. 5
In addition, several leading financial investment firms, particularly those that managed and sold the high-risk, mortgage-backed financial products, failed quickly because they had not set aside enough money to cover the billions of dollars they lost on mortgages now going into default. For example, the venerable financial company Bear Stearns, which had been a successful business for more than 85 years, was eventually sold to JP Morgan for less than $10 a share, even after the Federal Reserve made more than $50 billion dollars available to help prop up financial institutions in trouble. 6
After the collapse of Bear Stearns and other firms such as Lehman Brothers and insurance giant AIG, the Fed set up a special loan program to stabilize the banking system and to keep the U.S. bond markets trading at a normal pace. It is estimated that the Federal Reserve made more than $9 trillion in loans to major banks and other financial firms during the two-year crisis—not to mention bailing out the auto industry and buying several other firms to keep the financial system afloat. 7
As a result of this financial meltdown, Congress passed legislation in 2010 to implement major regulations in the financial industry to prevent the future collapse of financial institutions, as well to put a check on abusive lending practices by banks and other firms. Among its provisions, the Dodd-Frank Wall Street Reform and Consumer Protection Act (known as Dodd-Frank) created an oversight council to monitor risks that affect the financial industry; requires banks to increase their cash reserves if the council feels the bank has too much risk in its current operations; prohibits banks from owning, investing, or sponsoring hedge funds, private equity funds, or other proprietary trading operations for profit; and set up a whistle-blower program to reward people who come forward to report security and other financial violations. 8
Another provision of Dodd-Frank legislation requires major U.S. banks to submit to annual stress tests conducted by the Federal Reserve. These annual checkups determine whether banks have enough capital to survive economic turbulence in the financial system and whether the institutions can identify and measure risk as part of their capital plan to pay dividends or buy back shares. In 2017, seven years after Dodd-Frank became law, all of the country’s major banks passed the annual examination. 9
CONCEPT CHECK
- What are the four key functions of the Federal Reserve System?
- What three tools does the Federal Reserve System use to manage the money supply, and how does each affect economic activity?
- What was the Fed’s role in keeping the U.S. financial markets solvent during the 2007–2009 financial crisis?


1.3: Cardiovascular Anatomy
- Identify cardiovascular system gross anatomical structures.
- Identify cardiovascular system microscopic structures.
Gross and Microscopic Anatomy
Use local resources to view structures of the cardiovascular system.
Circulatory System Structures List
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Circulatory System- Arteries- Head and shoulders
- Aortic arch
- Brachiocephalic (brachiocephalic trunk)
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Common carotid arteries
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External carotid artery
- Facial artery
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External carotid artery
- Subclavian arteries
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Circulatory System- Arteries- Trunk
- Descending aorta
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Celiac trunk
- Left gastric artery
- Splenic artery
- Common hepatic artery
- Superior mesenteric artery
- Renal artery
- Gonadal artery
- Inferior mesenteric artery
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Common iliac artery
- Internal iliac artery
- External iliac artery
- Median (medial) sacral artery
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Circulatory System- Arteries- Arms
- Axillary artery
- Subscapular artery
- Humeral circumflex artery
- Brachial artery
- Radial artery
- Ulnar artery
- Interosseous artery
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Circulatory System- Arteries- Hands and upper legs
- Superficial palmar arch
- Femoral artery
- Lateral circumflex artery
- Medial circumflex artery
- Deep femoral artery
- Descending genicular artery
- Popliteal artery
- Lateral superior genicular artery
- Medial superior genicular artery
- Lateral inferior genicular artery
- Medial inferior genicular artery
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Circulatory System- Arteries- Lower legs and feet
- Anterior tibial artery
- Dorsalis pedis artery
- Peroneal artery
- Posterior tibial artery
Heart Structures List
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Heart- Exterior- Anterior view
- Right atrium
- Right ventricle
- Left atrium
- Left ventricle
- Apex
- Base
- Superior vena cava
- Pulmonary trunk
- Pulmonary arteries
- Pulmonary veins
- Aorta
- Brachiocephalic trunk
- Anterior interventricular artery
- Great cardiac vein
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Heart- Exterior- Left lateral view
- Inferior vena cava
- Circumflex branch of left coronary artery
- Left marginal artery
- Left marginal vein
- Coronary sinus
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Heart- Exterior- Right lateral view
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Right coronary artery
- Muscular artery
- Right marginal artery
- Anterior cardiac vein
- Right marginal vein
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Right coronary artery
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Heart- Open atrium- Lateral view
- Pectinate muscle
- Fossa ovalis
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Heart- Open ventricles- Right anterior view
- Tricuspid valve
- Chordae tendineae
- Papillary muscle
- Trabeculae carneae
- Pulmonary semilunar valve
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Heart- Open ventricles- Left anterior view
- Interventricular septum
- Bicuspid valve
- Aortic semilunar valve
Attributions for chapter 3:
Circulatory System Structures List by Marissa Sumida / CC BY 4.0
Heart Structures List by Marissa Sumida / CC BY 4.0


12.2: “Vectorized” Arithmetic Operators
Recall our table of Python math operators (Figure 5.1.1). What do those things do if we use them on aggregate, instead of atomic data? The answer is: something super cool and useful.
Operating on an Array and a Single Value
Consider the following code:
Code \(\PageIndex{1}\) (Python):
num_likes_today = np.array([6,61,0,0,14])
num_likes_tomorrow = num_likes_today + 3
print(num_likes_tomorrow)
| [ 9 64 3 3 17 ]
See what happened? “Adding 3” to the array means adding 3 to each element. All in one compact line of code, we can do five – or even five billion – operations. This works for all the other Figure 5.1.1 operators as well.
For somewhat geeky reasons, this sort of thing is called a vectorized operation. All you need to know is that this means fast . And that’s “fast” in two different ways: fast to write the code (since instead of using a loop , which we’ll cover in 14, you just write a single statement with + and = signs), and more importantly, fast to execute. For more geeky reasons, the above code will run lightning fast even if num_likes_today had five hundred million elements instead of just five. As you’ll learn if you ever try it, a Python loop is much slower. 2
Don’t get me wrong: there are times we’ll have to use a loop because we have no choice. But the general rule with Python is: if you can figure out how to perform a calculation without using a loop, always do it!
Operating on Two Arrays
Possibly even cooler, we can even “+” (or “-”, or “*”, or...) two entire arrays together. Example:
Code \(\PageIndex{2}\) (Python):
salaries = np.array([38000, 102000, 55750, 29500, 250000])
raises = np.array([1000, 4000, 2000, 1000, 2000])
salaries = salaries + raises
print(salaries)
This code produces:
| [ 39000<PHONE_NUMBER>0 30500 252000]
Can you see why? “Adding” the two arrays together performed addition element-by-element . The result is a new array with 38000+ 1000 as the first element, 102000 + 4000 as the second, etc. This, too, is a lightning-fast, vectorized operation, and it too works with all the other math operators.
Just to re-emphasize one point before we go on. In the example back on p. 77, we assigned the result of the operation to a new variable, num_likes_tomorrow . This means that num_likes_today itself was unchanged by the code. In contrast, in the example we just did, we assigned the result of the operation back into an existing variable ( salaries ). So salaries has itself been updated as a result of that code.


5.2.3: Diatomic Molecules of the First and Second Periods
First Period Homonuclear Diatomic Molecules
In the first row of the periodic table, the valence atomic orbitals are \(1s\). There are two possible homonuclear diatomic molecules of the first period:
Dihydrogen, H 2 \([\sigma_g^2(1s)]\): This is the simplest diatomic molecule. It has only two molecular orbitals (\(\sigma_g\) and \(\sigma_u^{*}\)), two electrons, a bond order of 1, and is diamagnetic. Its bond length is 74 pm . MO theory would lead us to expect bond order to decrease and bond length to increase if we either add or subtract one electron. The calculated bond length for the \(\ce{H_2^{+}}\) ion is approximately 105 pm . 1
Dihelium, He 2 \([\sigma_g^2\sigma_u^{*2}(1s)]\): This molecule has a bond order of zero due to equal number of electrons in bonding and antibonding orbitals. Like other nobel gases, He exists in the atomic form and does not form bonds at ordinary temperatures and pressures.
Draw the complete molecular orbital diagrams for \(\ce{H2}\) and for \(\ce{He2}\). Include sketches of the atomic and molecular orbitals.
- Answer
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A complete molecular orbital diagram includes all atomic orbitals and molecular orbitals, their symmetry labels, and electron filling.
Second Period Homonuclear Diatomic Molecules
The second period elements span from Li to Ne. The valence orbitals are 2s and 2p. In their molecular orbital diagrams, non-valence orbitals (1s in this case) are often disregarded in molecular orbital diagrams.
Orbital mixing has significant consequences for the magnetic and spectroscopic properties of second period homonuclear diatomic molecules because it affects the order of filling of the \(\sigma_g(2p)\) and \(\pi_u(2p)\) orbitals. Early in period 2 (up to and including nitrogen), the \(\pi_u(2p)\) orbitals are lower in energy than the \(\sigma_g(2p)\) (see Figure \(\PageIndex{1})\). However, later in period 2, the \(\sigma_g(2p)\) orbitals are pulled to a lower energy. This lowering in energy of \(\sigma_g(2p)\) is not unique; all of the \(\sigma\) orbitals in the molecule are pulled to lower energy due to the increasing positive charge of the nucleus. The \(\pi\) orbitals in the molecule are also affected, but to a much lesser extent than \(\sigma\) orbitals. The reason has to do with the high penetration of \(s\) atomic orbitals compared to \(p\) atomic orbitals (recall our previous discussion on penetration and shielding , and its effect on periodic trends ). The \(\sigma\) molecular orbitals have more \(s\) character and thus their energy is more influenced by increasing nuclear charge. As nuclear charge increases, the energy of the \(\sigma_g(2p)\) orbital is lowered significantly more than the energy of the \(\pi_u(2p)\) orbitals (Figure \(\PageIndex{1})\).
Dilithium, Li 2 \([\sigma_g^2(2s)]\): This molecule has a bond order of one and is observed experimentally in the gas phase to have one Li-Li bond.
Diberylium, Be 2 \([\sigma_g^2\sigma_u^{*2}(2s)]\): This molecule has a bond order of zero due to the equal number of electrons in bonding and antibonding orbitals. Although Be 2 does not exist under ordinary conditions, it can be produced in a laboratory and its bond length measured (Figure \(\PageIndex{2}\)). Although the bond is very weak, its bond length is surprisingly ordinary for a covalent bond of the second period elements. 2
Diboron, B 2 \([\sigma_g^2\sigma_u^{*2}(2s)\pi_u^1\pi_u^1(2p)]\): The case of diboron is one that is much better described by molecular orbital theory than by Lewis structures or valence bond theory. This molecule has a bond order of one. The molecular orbital description of diboron also predicts, accurately, that diboron is paramagnetic. The paramagnetism is a consequence of orbital mixing, resulting in the \(\sigma_g\) orbital's being at a higher energy than the two degenerate \(\pi_u^*\) orbitals.
Dicarbon, C 2 \([\sigma_g^2\sigma_u^{*2}(2s)\pi_u^2\pi_u^2(2p)]\): This molecule has a bond order of two. Molecular orbital theory predicts two bonds with \(\pi\) symmetry, and no \(\sigma\) bonding. C 2 is rare in nature because its allotrope, diamond, is much more stable.
Dinitrogen, N 2 \([\sigma_g^2\sigma_u^{*2}(2s)\pi_u^2\pi_u^2\sigma_g^2(2p)]\): This molecule is predicted to have a triple bond. This prediction is consistent with its short bond length and bond dissociation energy. The energies of the \(\sigma_g(2p)\) and \(\pi_u(2p)\) orbitals are very close, and their relative energy levels have been a subject of some debate (see next section for discussion).
Dioxygen, O 2 \([\sigma_g^2\sigma_u^{*2}(2s)\sigma_g^2\pi_u^2\pi_u^2\pi_g^{*1}\pi_g^{*1}(2p)]\): This is another case where valence bond theory fails to predict actual properties. Molecular orbital theory correctly predicts that dioxygen is paramagnetic, with a bond order of two. Here, the molecular orbital diagram returns to its "normal" order of orbitals where orbital mixing could be somewhat ignored, and where \(\sigma_g(2p)\) is lower in energy than \(\pi_u(2p)\).
Difluorine, F 2 \([\sigma_g^2\sigma_u^{*2}(2s)\sigma_g^2\pi_u^2\pi_u^2\pi_g^{*2}\pi_g^{*2}(2p)]\): This molecule has a bond order of one and like oxygen, the \(\sigma_g(2p)\) is lower in energy than \(\pi_u(2p)\).
Dineon, Ne 2 \([\sigma_g^2\sigma_u^{*2}(2s)\sigma_g^2\pi_u^2\pi_u^2\pi_g^{*2}\pi_g^{*2}\sigma_u^{*2}(2p)]\): Like other noble gases, Ne exists in the atomic form and does not form bonds at ordinary temperatures and pressures. Like Be 2 , Ne 2 is an unstable species that has been created in extreme laboratory conditions and its bond length has been measured (Figure \(\PageIndex{2}\))
Draw the complete molecular orbital diagram for O 2 . Show calculation of its bond order and tell whether it is diamagnetic or paramagnetic.
- Answer
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O 2 is paramagnetic with a bond order of 2. Its \(\sigma_g(2p)\) molecular orbital is lower in energy that the set of \(\pi_{u}(2p)\) orbitals.
Bond order \(=\frac{1}{2}\left[\left(\begin{array}{c}\text { 8 electrons in} \\ \text { valence bonding orbitals }\end{array}\right)-\left(\begin{array}{c}\text {4 electrons in} \\ \text { valence antibonding orbitals }\end{array}\right)\right]\)
Use a qualitative molecular orbital energy-level diagram to predict the electron configuration, the bond order, and the number of unpaired electrons in the peroxide ion (O 2 2 − ).
- Answer
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This diagram looks similar to that of \(\ce{O2}\), except that there are two additional electrons.
\( \left ( \sigma _{g}(2s) \right )^{2}\left ( \sigma_u ^{\star }(2s) \right )^{2}\left ( \sigma _g(2p) \right )^{2}\left ( \pi _{u}(2p) \right )^{4}\left ( \pi _g(2p) \right )^{4} \); bond order of 1; no unpaired electrons.
Bond Lengths in Homonuclear Diatomic Molecules
The trends in experimental bond lengths are predicted by molecular orbital theory, specifically by the calculated bond order. The values of bond order and experimental bond lengths for the second period diatomic molecules are given in Figure \(\PageIndex{1}\), and shown in graphical format on the plot in Figure \(\PageIndex{2}\). From the plot, we can see that bond length correlates well with bond order, with a minimum bond length occurring where the bond order is greatest (\(\ce{N2}\)). The shortest bond distance is at \(\ce{N2}\) due to its high bond order of 3. From \(\ce{N2}\) to \(\ce{F2}\) the bond distance increases despite the fact that atomic radius decreases.
References
- NIST, Calculated Geometries available for H 2 + (Hydrogen cation) 2Σg+ D∞h, available at https://cccbdb.nist.gov/diatomicexpbondx.asp
- Merritt, J. M.; Bondybey, V. E.; Heaven, M. C., Beryllium Dimer-Caught in the Act of Bonding. Science 2009, 324 (5934), 1548-1551.


3: Membrane Phases and Morphologies
One of the most important properties of a lipid bilayer is the relative mobility (fluidity) of the individual lipid molecules and how this mobility changes with temperature. This response is known as the phase behavior of the bilayer. The phase behavior of lipid bilayers is largely determined by the strength of the attractive Van der Waals interactions between adjacent lipid molecules. The extent of this interaction is in turn governed by how long the lipid tails are and how well they can pack together.
-
- 3.1: Membrane Phase Transitions
- Biological membranes are primarily composed of phospholipids—a diverse class of compounds composed of a hydrophilic head group covalently attached to a pair of hydrophobic fatty acids. This amphipathic structure leads phospholipid molecules to spontaneously form bilayers when placed in water, as the phospholipids are driven to orient their head groups towards water and shield their fatty acid tails from it via the hydrophobic effect.
-
- 3.2: The Main Phase Transition
- The membranes that provide structure and definition to cellular compartments are composed of dynamic and heterogeneous lipids whose varying chemical properties allow for their function. The transition between the two main phases of a lipid bilayer, the liquid crystalline phase (Lα) and the gel phase (Lβ) determines how the membrane will behave in terms of fluidity at a particular temperature and the formation of biologically useful membrane microdomains.
-
- 3.4: The Gel Phase
- Biological plasma membranes express a plethora of unique lipids, sterols, and proteins and exhibit greater complexity than model lipid bilayer systems. At temperatures above the melting point the lipid bilayer exists in the liquid phase and at lower temperatures, the lipid bilayer enters the gel phase. In the gel phase, the lipid bilayer undergoes a fundamental molecular reorganization characterized by an increase in order, a loss of lateral mobility and a change in the broad topology of the mem
-
- 3.5: The Ripple Phase
- This phase is characterized by corrugations of the membrane surface with well-defined periodicity with an axis parallel to the mean bilayer plane [1]. The molecular origin of ripple-phase formation is traditionally been associated with the lipid headgroup region and hence lipids can be classified into ripple-forming and non-ripple forming lipids based on their headgroups.
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- 3.6: Rafts
- Lipid “rafts” are membrane microdomains composed of higher order lipids that have tight interaction with each other relative to the “sea” of liquid phase lipids known as bulk lipids. These rafts have been estimated to be 10-200 nm in size. Small rafts can merge to form large rafts through protein-protein and protein-lipid interactions. The rafts are important because certain membrane proteins preferentially localize to these domains and biological processes such as signal transduction occur ther


17: Circulatory and Lymphatic System Infections (Part C)
Yellow fever was once common in the southeastern US, with annual outbreaks of more than 25,000 infections in New Orleans in the mid-1800s. [1] In the early 20th century, efforts to eradicate the virus that causes yellow fever were successful thanks to vaccination programs and effective control (mainly through the insecticide dichlorodiphenyltrichloroethane [DDT]) of Aedes aegypti , the mosquito that serves as a vector. Today, the virus has been largely eradicated in North America.
Elsewhere, efforts to contain yellow fever have been less successful. Despite mass vaccination campaigns in some regions, the risk for yellow fever epidemics is rising in dense urban cities in Africa and South America. [2] In an increasingly globalized society, yellow fever could easily make a comeback in North America, where A. aegypti is still present. If these mosquitoes were exposed to infected individuals, new outbreaks would be possible.
Like yellow fever, many of the circulatory and lymphatic diseases discussed in this chapter are emerging or re-emerging worldwide. Despite medical advances, diseases like malaria, Ebola, and others could become endemic in the US given the right circumstances.
Chapter Outline
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- 17.1: Viral Infections of the Circulatory and Lymphatic Systems
- Viral pathogens of the circulatory system vary tremendously both in their virulence and distribution worldwide. Some of these pathogens are practically global in their distribution. Fortunately, the most ubiquitous viruses tend to produce the mildest forms of disease. In the majority of cases, those infected remain asymptomatic. On the other hand, other viruses are associated with life-threatening diseases that have impacted human history.
Footnotes
- Centers for Disease Control and Prevention. “The History of Yellow Fever.” http://www.cdc.gov/travel-training/local/HistoryEpidemiologyandVaccination/page27568.html
- C.L. Gardner, K.D. Ryman. “Yellow Fever: A Reemerging Threat.” Clinical Laboratory Medicine 30 no. 1 (2010):237–260.


12.1: Harvesting Workflows
Development of content in the LibreTexts platform proceeds via five mechanisms operating in parallel:
- Mechanism 1: Student contribution from scratch (e.g., via extra credit)
- Mechanism 2 : Student integration of existing content outside the LibreTexts platform from faculty and experts
- Mechanism 3: Faculty construction of raw content from scratch
- Mechanism 4: Faculty integration of existing content outside the LibreTexts platform from faculty and experts
- Mechanism 5: Faculty and student re-editing of content already existing on the LibreTexts (with permission)
Modules from all five mechanisms are processed through a sophisticated vetting structure, involving students and faculty (see below), to eventually guarantee reliable, fully-vetted content that is capable of substituting for that found in current paper-based textbooks. Content from higher ranked mechanisms typically supersedes content from lower ranked mechanisms, resulting in the continual evolution of the LibreTexts content.
There is a range of different workflows possible for constructing texts and selecting the best workflow is an individualize or team decision. There are tools to facilitate construction efforts with the Remixer as most powerful.
Harvesting project follow different workflows depending on the nature of the content and the nature of the format of the content. Below are several established workflows that may be followed in harvesting projects.
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From Scratch
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From Website (online)
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From PDF Mechanism I
Mechanism II
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From Word Mechanism I
Mechanism II
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From Google Doc Mechanism I
Mechanism II
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From EPUB/Common Cartridge
If the EPUB is unvalidated (poorly constructed), then the LT importer will not generate all pages and will push chapter content into single pages. If this happened, then user Remixer (in remix mode) to flesh out the text generated from the Importer (e.g. add relevant pages). |
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From Latex Source
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From Pressbooks Mechanism I (if EPUB export option is available)
Mechanism II (if EPUB export option is not available)
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From Lumen (modified Pressbooks) Lumen platform is really a Pressbooks platform slightly modified with no export options (OER content is fettered).
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From EPUB3 Direct Importer under development
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From XML Direct Importer under development |
From Pretexts Requires admin access to the importer to harvest. |


12: Protein Metabolism of Unknowns
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- Valeria Hochman Adler
- Reedley College
Learning Outcomes
By the end of this lab period, you will be able to:
- Describe the pattern of protein metabolism for each of your organisms for the tests that we completed
- Articulate how pH indicators such as phenol red, bromocresol purple, phenol red, etc. are used to track protein utilization.
- Explain how differences in the expression of specific enzymes (such as nitrate reductase, gelatinase, and urease, etc.) can be used to differentiate between bacteria species.
Introduction
In today’s lab, we will be continuing with a series of biochemical tests that will help you further narrow down which organisms are your Gram-positive and Gram-negative unknowns. The focus of today’s tests is on the metabolism and breakdown of proteins, and the metabolism of nitrogen.
The tests that we will be setting up today are
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Decarboxylase tests
- Lysine
- Ornithine
- Phenylalanine deaminase
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SIM
- Sulfur
- Motility
- Indole
- Nitrate reduction
- Gelatin hydrolysis
- Urea hydrolysis
Decarboxylase Tests
Amino acids are structurally composed of an amino group (NH 2 ), a “carboxy” or carboxylic acid group (COOH), and an “R” group, which is the part of the molecule that is variable. Enzymes that remove the carboxylic acid group are called “decarboxylases”. Determining whether or not a bacteria makes a specific decarboxylase can be useful in their identification. These tests are useful for helping us ID our Gram-negative bacteria, but not helpful for Gram-positive, so well only perform decarboxylase tests on our Gram-negative.
Today, we will be testing for two decarboxylase enzymes, ornithine decarboxylase, and lysine decarboxylase. Decarboxylase broths contain the pH indicator bromocresol purple which turns yellow in acidic conditions, and purple in basic solutions. After inoculation, a layer of mineral oil is overlaid onto the broth, creating anaerobic conditions and inducing the bacteria to ferment glucose. Initially, this will result in the production of acid and the media will turn yellow. However, for those organisms that produce the decarboxylase enzyme, the presence of the amino acid in the media will turn “on” the gene for the enzyme, and result in the production of amines. These amines accumulate and create a basic environment turning the media purple. If no decarboxylase is produced, the medium will remain yellow.
Phenylalanine Deaminase
In addition to enzymes that remove the carboxylic acid group, some bacteria have enzymes that can remove the amino group from specific amino acids. Determining if an organism can remove the amino group from phenylalanine is also a useful test for identifying Gram-negative organisms.
The phenylalanine deaminase test is done by inoculating a phenylalanine slant. If the bacteria make the enzyme, it will react with the phenylalanine and produce phenylpyruvic acid. Phenylpyruvic acid is colorless but will react with ferric chloride (FeCl 3 ) to produce a forest green color.
SIM Tests
The SIM tests, again only performed on our Gram-negative bacteria, are actually three tests in one tube. In this tube, we will test for s ulfur reduction, the production of i ndole, and bacterial m otility. Both sulfur and indole production are the result of amino acid metabolism. The observation of motility is just a happy “extra” that we can often glean from this test.
SIMs media is a type of “deep” and is inoculated with a needle.
Sulfur reduction
The amino acid cysteine contains a sulfur atom in its “R” group. Some bacteria make an enzyme called cysteine desulfurase, which removes that sulfur atom in the form of hydrogen sulfide gas. The hydrogen sulfide gas will react with ferrous ammonium sulfate (FeSO 4 ) to produce a black precipitate, ferric sulfide (FeS).
There are other mechanisms by which bacteria can reduce sulfur. The enzyme thiosulfate reductase can reduce sodium thiosulfate to form hydrogen gas as well. This will also result in the formation of a black precipitate.
Indole
The enzyme tryptophanase hydrolyzes the amino acid tryptophan to pyruvate, ammonia, and indole. Indole will react with p-dimethylamino benzaldehyde (a component of Kovak’s reagent) to form a cherry red color, detected as a layer on top of the deep.
Motility
It is frequently serendipitous that we can determine motility from SIMs media. However, it is only possible to do this when sulfur is not reduced and the tube does not turn black. Because H2S gas can diffuse rapidly in the media, its presence usually results in a blackening of the entire tube, obscuring any motility observation.
If your organism is negative for sulfur reduction, hold the tube up to the light and look for growth that radiates away from the initial inoculation. This indicates that the bacteria are able to move through the media. If the media in the tube becomes cloudy as compared to an uninoculated control, this indicates that the organism is motile.
Nitrate Reduction Test
Many bacteria make an enzyme nitrate reductase that can reduce nitrate (NO 3 ) to nitrite (NO 2 ), while other organisms can reduce nitrate (NO 3 ) further, producing a set of possible intermediates and final products including ammonia (NH 3 ) and nitrogen gas (N 2 ).
We identify organisms that have reduced nitrate to nitrite by the addition of two reagents (Nitrate Reagent A, and Nitrate Reagent B). If nitrite has been produced the nitrate broth will turn a deep blood red. That is a positive result and can be recorded.
However, because some organisms further reduce nitrite, it’s possible that your organism DID reduce nitrate, but there is simply no nitrite left in the broth because it has all been converted to any of the further endproducts or their intermediates shown in Figure \(\PageIndex{6}\). So how can you tell whether there was NO nitrate reduction, or that reduction continued past the production of nitrite? We do this by adding some powdered zinc to the nitrate broth. Zinc will react with any remaining nitrate, converting it into nitrite which will then react with Reagents A and B that you added previously. If the tube turns red after the addition of zinc, this is a negative result. If the tube does not turn red, then there was neither nitrate nor nitrite left in the tube, and this would be recorded as positive for nitrate reduction.
Gelatin Hydrolysis
Earlier in the semester when we learned about agar, we discussed how gelatin was once used (or its use was attempted) to create solid substrates for growing bacteria. We learned that gelatin is a terrible choice because so many bacteria secrete enzymes called gelatinases that break down and liquefies the gelatin. The gelatin hydrolysis test differentiates between organisms that produce gelatinases, and those that do not. The gelatin test is performed in a “deep” and is inoculated with an inoculating needle.
Urea Hydrolysis
A common byproduct of the decarboxylation of amino acids is the urea molecule. Urea is a metabolic waste product, but the nitrogen in urea can be recycled by some microbes that make the enzyme urease. This enzyme is also a virulence factor in the pathogen Helicobacter pylori - it uses this enzyme to generate urea which neutralizes the acid in your stomach where it lives and reproduces.
Urea broth contains only a tiny amount of yeast extract for nutrition and strong buffers that prevent a color change from occurring unless enough ammonia is rapidly produced. Phenol red is the pH indicator, which is pink above pH 8.4. An orange or yellow result is negative.
Materials
Day 1
- One Lysine decarboxylase broth (for Gram-negative only)
- One Ornithine decarboxylase broth (for Gram-negative only)
- Sterile mineral oil in microcentrifuge tubes
- One Phenylalanine deaminase slant (for Gram-negative only)
- One SIM deep (for Gram-negative only)
- Two nitrate broths
- Two nutrient gelatin deeps
- Two urea broths
Day 2
- 12% ferric chloride solution
- Kovac’s reagent
- Nitrate test reagents A and B
- Zinc powder
- Ice
Experiment
Day 1
- Inoculate the lysine decarboxylase and ornithine decarboxylase broths with your Gram-negative organism. Overlay both tubes with 3 to 4 mm sterile mineral oil. Incubate at 37 o C for 96 hours (up to one week).
- Inoculate the phenylalanine deaminase slant with your Gram-negative organism.
- Use an inoculation needle to inoculate a SIM tube with your Gram-negative organism. Insert the needle into the agar to within 1 cm of the bottom of the tube. Be careful to remove the needle along the original insertion line.
- Inoculate 2 nitrate broths, one with your Gram-positive and one with your Gram-negative organism.
- Use an inoculation needle to stab inoculate the nutrient gelatin medium.
- Inoculate 2 urea broths, one with your Gram-positive and one with your Gram-negative organism.
- Place all freshly inoculated cultures in the 37 o C incubator, and incubate for 48 hours. If this is a 5-day incubation, incubate at 30 o C to avoid cultures growing past stationary phase into death phase.
- Each week during the ID project you should be making fresh subcultures of your organisms and verifying purity by gram staining. Depending on how old your TSA slants are this can be done on Day 1 OR Day 2.
Day 2
- Remove all of your cultures from the incubator. Carefully inspect each decarboxylase broth, and note whether the result was positive (purple), negative (yellow or no change of color), or indeterminate. Compare the colors with the uninoculated control tubes. Sometimes a result is weakly positive - make a note of this! Decarboxylation tests may need to be incubated for up to one week. Return the broth tubes back to the 37 o C incubator, and make one more observation in the next lab period if the tube is yellow or indeterminate.
- For the phenylalanine deaminase test, add a few drops of 12% ferric chloride solution to the slant and observe for color change (green). (Note: this color may fade quickly, so record your results immediately).
- For the SIM test, examine the tube for spreading from the stab line and formation of black precipitate in the medium. Record any H 2 S production and/or motility. Add Kovac’s reagent to each tube to a depth of 2 mm - 3 mm. After several minutes, observe the formation of red color in the reagent layer and record the Indole result.
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For the nitrate reduction test, examine each tube for evidence of gas production. Record your results. Proceed to the addition of reagents to all tubes as follows:
- Add 8 drops of reagent A and 8 drops of reagent B to each tube. Mix well, but carefully, and let the tubes stand undisturbed for 10 minutes. Record the tube that has red color as nitrate reduction-positive.
- For any tube that did not change color, add a pinch of zinc dust. Mix well, and let the tubes stand for 10 minutes. If the color turns red, record the organism as nitrate reduction-negative. If there is no color change after adding zinc, record the organism as nitrate reduction-positive.
- For the gelatin hydrolysis test, place all tubes in the ice bath until the control tube is solid. Examine the inoculated medium for gelatin liquefaction. Tip over the tubes to take photos of your test tubes, be careful not to spill liquefied gelatin out the top. Record your results. The gelatin test may need to be incubated for up to one week. Keep the gelatin tubes, and make one more observation in the next lab period.
- Observe the color of the urease test tubes, and record any tube that has turned pink as urease-positive.
Data
Your primary goal as you leave lab today is to have a complete and well-documented record of the results you collected today. Be sure to update your test table with the new data! Continue to develop your dichotomous keys and identify your unknowns.


12: Collaboration In And Across Teams Last updated Save as PDF Page ID 46040 Nina Burokas Lumen Learning Learning Objectives Discuss the role and types of teams in workplace communication Discuss strategies for working in collaborative projects Discuss best practices in workplace etiquette 12.1: Collaborative Writing 12.2: Introduction to Workplace Etiquette 12.3: Interaction in the Workplace 12.4: Company Events 12.5: Putting It Together- Collaboration in and across teams 12.6: Assignment- Collaboration in and Across Teams 12.7: Why It Matters- Collaboration In And Between Teams 12.8: Introduction to Team Communication in the Workplace 12.9: The Value of Teams in Business Communication 12.10: Teams and Team Dynamics 12.11: Group Communication Networks 12.12: Conflict Resolution 12.13: Introduction to Collaborative Projects 12.14: Decision Making In Groups 12.15: Tools for Communicating with Groups


11.4: The Impacts of Earthquakes
Some of the common impacts of earthquakes include structural damage to buildings, fires, damage to bridges, highways, pipelines and electrical transmission lines, initiation of slope failures, liquefaction, and tsunami. The types of impacts depend to a large degree on where the earthquake is located: whether it is predominantly urban or rural, densely or sparsely populated, highly developed or underdeveloped, and of course on the ability of the infrastructure to withstand shaking.
As we’ve seen from the example of the 1985 Mexico earthquake, the geological foundations on which structures are built can have a significant impact on earthquake shaking. When an earthquake happens, the seismic waves produced have a wide range of frequencies. The energy of the higher frequency waves tends to be absorbed by solid rock, while the lower frequency waves (with periods slower than one second) pass through the solid rock without being absorbed, but are eventually absorbed and amplified by soft sediments. It is therefore very common to see much worse earthquake damage in areas underlain by soft sediments than in areas of solid rock. A good example of this is in the Oakland area near San Francisco, where parts of a two-layer highway built on soft sediments collapsed during the 1989 Loma Prieta earthquake (Figure \(\PageIndex{1}\)).
Building damage is also greatest in areas of soft sediments, and multi-storey buildings tend to be more seriously damaged than smaller ones. Buildings can be designed to withstand most earthquakes, and this practice is increasingly applied in earthquake-prone regions. Turkey is one such region, and even though Turkey had a relatively strong building code in the 1990s, adherence to the code was poor, as builders did whatever they could to save costs, including using inappropriate materials in concrete and reducing the amount of steel reinforcing. The result was that there were over 17,000 deaths in the 1999 M7.6 Izmit earthquake (Figure \(\PageIndex{2}\)). After two devastating earthquakes that year, Turkish authorities strengthened the building code further, but the new code has been applied only in a few regions, and enforcement of the code is still weak, as revealed by the amount of damage from a M7.1 earthquake in eastern Turkey in 2011.
Fires are commonly associated with earthquakes because fuel pipelines rupture and electrical lines are damaged when the ground shakes (Figure \(\PageIndex{3}\)). Most of the damage in the great 1906 San Francisco earthquake was caused by massive fires in the downtown area of the city (Figure \(\PageIndex{4}\)). Some 25,000 buildings were destroyed by those fires, which were fuelled by broken gas pipes. Fighting the fires was difficult because water mains had also ruptured. The risk of fires can be reduced through P-wave early warning systems if utility operators can reduce pipeline pressure and close electrical circuits.
Earthquakes are important triggers for failures on slopes that are already weak. An example is the Las Colinas slide in the city of Santa Tecla, El Salvador, which was triggered by a M7.6 offshore earthquake in January 2001 (Figure \(\PageIndex{5}\)).
Ground shaking during an earthquake can be enough to weaken rock and unconsolidated materials to the point of failure, but in many cases the shaking also contributes to a process known as liquefaction , in which an otherwise solid body of sediment is transformed into a liquid mass that can flow. When water-saturated sediments are shaken, the grains become rearranged to the point where they are no longer supporting one another. Instead, the water between the grains is holding them apart and the material can flow. Liquefaction can lead to the collapse of buildings and other structures that might be otherwise undamaged. A good example is the collapse of apartment buildings during the 1964 Niigata earthquake (M7.6) in Japan (Figure \(\PageIndex{6}\)). Liquefaction can also contribute to slope failures and to fountains of sandy mud (sand volcanoes) in areas where there is loose saturated sand beneath a layer of more cohesive clay.
Parts of the Fraser River delta are prone to liquefaction-related damage because the region is characterized by a 2 metre to 3 metre thick layer of fluvial silt and clay over top of at least 10 meters of water-saturated fluvial sand (Figure \(\PageIndex{7}\)). Under these conditions, it can be expected that seismic shaking will be amplified and that the sandy sediments will liquefy. This could lead to subsidence and tilting of buildings, and to failure and sliding of the silt and clay layer. Current building-code regulations in the Fraser delta area require that measures be taken to strengthen the ground underneath multi-storey buildings prior to construction.
There are a few ways that you can demonstrate the process of liquefaction for yourself. The simplest is to go to a sandy beach (lake, ocean, or river) and find a place near the water’s edge where the sand is wet. This is best done with your shoes off, so let’s hope it’s not too cold! While standing in one place on a wet part of the beach, start moving your feet up and down at a frequency of about once per second. Within a few seconds the previously firm sand will start to lose strength, and you’ll gradually sink in up to your ankles.
If you can’t get to a beach, or if the weather isn’t cooperating, put some sand (sandbox sand will do) into a small container, saturate it with water, and then pour the excess water off. You can shake it gently to get the water to separate and then pour the excess water away, and you may have to do that more than once. Place a small rock on the surface of the sand; it should sit there for hours without sinking in. Now, holding the container in one hand gently thump the side or the bottom with your other hand, about twice a second. The rock should gradually sink in as the sand around it becomes liquefied.
As you were moving your feet up and down or thumping the pot, it’s likely that you soon discovered the most effective rate for getting the sand to liquefy; this would have been close to the natural harmonic frequency for that body of material. Stepping up and down as fast as you can (several times per second) on the wet beach would not have been effective, nor would you have achieved much by stepping once every several seconds. The body of sand vibrates most readily in response to shaking that is close to its natural harmonic frequency, and liquefaction is also most likely to occur at that frequency.
See Appendix 3 for Exercise 11.4 answers .
Earthquakes that take place beneath the ocean have the potential to generate tsunami . (Tsunami is the Japanese word for harbour wave. It is the same in both singular and plural.) The most likely situation for a significant tsunami is a large (M7 or greater) subduction-related earthquake. As shown in Figure \(\PageIndex{9}\), during the time between earthquakes the overriding plate becomes distorted by elastic deformation; it is squeezed laterally (Figure \(\PageIndex{9}\)B) and pushed up.
When an earthquake happens (Figure \(\PageIndex{9}\)C), the plate rebounds and there is both uplift and subsidence on the sea floor, in some cases by as much as several meters vertically over an area of thousands of square kilometers. This vertical motion is transmitted through the water column where it generates a series of waves that then spread across the ocean.
Subduction earthquakes with magnitude less than 7 do not typically generate significant tsunami because the amount of vertical displacement of the sea floor is minimal. Sea-floor transform earthquakes, even large ones (M7 to M8), don’t typically generate tsunami either, because the motion is mostly side to side, not vertical.
Tsunami waves travel at velocities of several hundred kilometers per hour and easily make it to the far side of an ocean in about the same time as a passenger jet. The simulated one shown in Figure \(\PageIndex{10}\) is similar to that created by the 1700 Cascadia earthquake off the coast of British Columbia, Washington, and Oregon, which was recorded in Japan nine hours later.
Tsunami are discussed further in Chapter 17 under the topic of waves and coasts.
Media Attributions
- Figure \(\PageIndex{1}\): “Cypress Collapsed” by the USGS. Public domain.
- Figure \(\PageIndex{2}\): “ Izmit, Turkey Earthquake ” by the USGS. Public domain.
- Figure \(\PageIndex{3}\): “SH-60B helicopter flies over Sendai” by the United States Navy. Public domain.
- Figure \(\PageIndex{4}\): “San Francisco Fire 1906” from the Library of Congress. Public domain.
- Figure \(\PageIndex{5}\): “ElSalvadorslide” by the USGS. Public domain.
- Figure \(\PageIndex{6}\): “Liquefaction at Niigata.” Public domain.
- Figures 11.4.7, 11.4.8, 11.4.9: © Steven Earle. CC BY.
- Figure \(\PageIndex{10}\): “ Maximum Tsunami Amplitude ” from NOAA/PMEL/Center for Tsunami Research . Public domain.


10.6: Gendered Prison Experience
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Prisons are not fun. People do not want to be in prison (despite what some few and far between news stories might claim). The free meals, health care, and rooms are not luxurious; in fact, in most institutions human rights are violated with spoiled food, lack of yard time, deprivation of family visitations or phone calls, lack of safety or privacy, medical experimentation, deprivation of rehabilitative services, and outright abuse and sexual assault.
Sociologist Gresham Sykes (1958) noted that prison inmates suffer through a variety of serious hardships, which he referred to as the “pains of imprisonment.” 325 Pains of imprisonment include loss of freedom and independence, loss of important familial, personal, romantic, and/or sexual relationships, inability to access normal goods or services, and loss of personal security. Collectively, these hardships can lead inmates into depression and may make them more susceptible to what sociologist Erving Goffman called “mortification.”
One example of deprivation of protection within prison can be seen in the habitual occurrence of rape—including prison staff as well as inmates—within the confines of prison. Prison staff includes security staff, teachers and counselors, medical workers, contractors and even religious volunteers. Struckman-Johnson conducted the most comprehensive research to date on prisoner rape. 326 After surveying 1,800 inmates in Midwestern prisons, they found that one in five male prisoners have been coerced or pressured into sex, and one in ten has been raped. In one women's prison, more than a quarter of the inmates said they had been pressured into sex by guards.
Political activist, professor, academic, and author Angela Davis argues that while men constitute the vast majority of prisoners in the world, important aspects of the state punishment system are missed if it is assumed that women are marginal and thus undeserving of attention. 327 Further, according to Davis, because women make up a relatively small proportion of the whole prison population, the inattention given to female prisoners is frequently justified. Due to the late twentieth-century reforms which relied on a “separate but equal” model, demands for more repressive conditions in order to render women’s facilities “equal” to men’s resulted in harsher punishments and disciplinary actions in women’s prisons than were previously implemented. Women of color in particular are subject to regimes of punishment that differ significantly from those experiences by white women, including assigned chores, manual labor, and frequent unnecessary strip searches. “Sexual abuse, especially among women of color, has become an institutionalized component of punishment behind prison walls.” 328
Conditions of improper touching by persons of authority, sanctioned sexual harassment, and unnecessary strip searches exist in numerous women’s prisons across the country.   Two noted that IDOC staff had sexually assaulted them more than ten times. About 23% of the respondents stated that IDOC staff had offered money, food or privileges in exchange for sexual favors, and almost 10% noted that staff had done this to them on more than three occasions. About one in seven reported that IDOC staff had threatened loss of privileges, physical attack, or placement in isolation if the woman refused sex with the staff.
The violent sexualization of prison life within women’s institutions exposes ideologies of sexuality—and the intersection of race and sexuality—which have had a profound effect on the representations of and treatment received by women of color both within and upon release from prison. 331 Men of color experience a perilous continuity in the way they are treated in prison, where they are more likely to experience harsher forms of punishment such as solitary confinement whereas women of color are more likely than white women to experience sexual abuse within prison. 332
The state itself is directly implicated in the routinization of sexual abuse in women’s prisons and other forms of physical and mental abuse such as solitary confinement in men’s prisons, both in permitting such conditions that render individuals (especially those of color) vulnerable to explicit abuse carried out by guards and other prison staff and by incorporating into routine policy such practices as the strip search and solitary confinement. Being victimized will likely result in further social-psychological damage for the inhabitants experiencing the institutionalized racism and sexism. This results in their adjustment to society upon release being even further hindered, with one likely consequence being a return to crime and recidivating back into the prison structure. 333 334 335
While nearly 5,000 transgender people are incarcerated in state prisons, it's estimated only 15 cases (people) in which these prisoners were housed according to their lived gender. 336 Based on the available records (many records remain sealed depending on states' privacy laws) obtained from 45 states, just 13 transgender women are housed with women and two transgender men are housed with men. Thirty-five percent of transgender people who had spent time in prison in the previous year reported being sexually assaulted by staff or other inmates, according to a 2015 report by the Department of Justice. 337 When asked about the experiences surrounding their victimization by other inmates, 72% said they experienced force or threat of force and 29% said they were physically injured.
325 Sykes, G.M. 1958.
The Society of Captives: A Study of a Maximum Security Prison
. Princeton, NJ: Princeton University Press.
326 Struckman-Johnson, Cindy. 2006.
National Prison Rape Elimination
. University Press. South Dakota.
327 Davis, Angela Y. 2003.
Are Prisons Obsolete?
Page 77. New York: Seven Stories Press.
328 Ibid.
329 Human Rights Watch Report. 2001. “No Escape: Male Rape in U.S. Prisons.” Retrieved October 2, 2008 (
http://www.hrw.org/legacy/reports/2001/prison/
)
330 Mills, Alan, Margaret Byrne. 2004. Rape Crisis in Women's Prison. Chicago Press: Chicago.
331 Davis, Angela Y. 2003. Are Prisons Obsolete? New York: Seven Stories Press.
332 Poole, Eric and Robert Regoli. 1980. “Race, Institutional Rule Breaking, and Disciplinary Response.” Law and Society Review, 14:4, 931-946.
333 Bonta, J., P. Gendreau. 1990. “Reexaming the Cruel and Unusal Punishment of Prison Life.” Law and Human Behavior, 347-366.
334 Cohen, S., L. Taylor. 1972. Psychological Survival. Hammondsworth: Penguin
335 Day, Susie. 2001. “Cruel But Not Unusual: The Punishment of Women in U.S. Prisons.” Monthly Review. Retrieved April 29, 2006 (
http://www.monthlyreview.org/0701day.htm
)
336 Sosin, K. (2020). Trans, imprisoned — and trapped. NBC News.
https://www.nbcnews.com/feature/nbc-
out/transgender-women-are-nearly-always-incarcerated-men-s-putting-many-n1142436
337 Beck, A. (2015). PREA Data Collection Activities, 2015. U.S. Department of Justice.
https://bjs.ojp.gov/content/pub/pdf/pdca15.pdf


10.4: Test for Differences in Means- Assuming Equal Population Variances
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Typically we can never expect to know any of the population parameters, mean, proportion, or standard deviation. When testing hypotheses concerning differences in means we are faced with the difficulty of two unknown variances that play a critical role in the test statistic. We have been substituting the sample variances just as we did when testing hypotheses for a single mean. And as we did before, we used a Student's t to compensate for this lack of information on the population variance. There may be situations, however, when we do not know the population variances, but we can assume that the two populations have the same variance. If this is true then the pooled sample variance will be smaller than the individual sample variances. This will give more precise estimates and reduce the probability of discarding a good null. The null and alternative hypotheses remain the same, but the test statistic changes to:
\[t_{c}=\frac{\left(\overline{x}_{1}-\overline{x}_{2}\right)-\delta_{0}}{\sqrt{S^{2} p\left(\frac{1}{n_{1}}+\frac{1}{n_{2}}\right)}}\nonumber\]
where \(S_{p}^{2}\) is the pooled variance given by the formula:
\[S_{p}^{2}=\frac{\left(n_{1}-1\right) s_{2}^{1}+\left(n_{2}-1\right) s_{2}^{2}}{n_{1}+n_{2}-2}\nonumber\]
Example \(\PageIndex{1}\)
A drug trial is attempted using a real drug and a pill made of just sugar. 18 people are given the real drug in hopes of increasing the production of endorphins. The increase in endorphins is found to be on average 8 micrograms per person, and the sample standard deviation is 5.4 micrograms. 11 people are given the sugar pill, and their average endorphin increase is 4 micrograms with a standard deviation of 2.4. From previous research on endorphins it is determined that it can be assumed that the variances within the two samples can be assumed to be the same. Test at 5% to see if the population mean for the real drug had a significantly greater impact on the endorphins than the population mean with the sugar pill.
Solution
First we begin by designating one of the two groups Group 1 and the other Group 2. This will be needed to keep track of the null and alternative hypotheses. Let's set Group 1 as those who received the actual new medicine being tested and therefore Group 2 is those who received the sugar pill. We can now set up the null and alternative hypothesis as:
\[
\begin{array}{l}
H_0: \mu_1 \leq \mu_2 \\
H_1: \mu_1>\mu_2
\end{array}
\]
This is set up as a one-tailed test with the claim in the alternative hypothesis that the medicine will produce more endorphins than the sugar pill. We now calculate the test statistic which requires us to calculate the pooled variance, \(S_p^2\) using the formula above.
\[
t_c=\frac{\left(\bar{x}_1-\bar{x}_2\right)-\delta_0}{\sqrt{S_p^2\left(\frac{1}{n_1}+\frac{1}{n_2}\right)}}=\frac{(8-4)-0}{\sqrt{20.4933\left(\frac{1}{18}+\frac{1}{11}\right)}}=2.31
\]
\(\mathrm{t}_\alpha\), allows us to compare the test statistic and the critical value.
\[
t_\alpha=1.703 \text { at } d f=n_1+n_2-2=18+11-2=27
\]
The test statistic is clearly in the tail, 2.31 is larger than the critical value of 1.703 , and therefore we cannot maintain the null hypothesis. Thus, we conclude that there is significant evidence at the \(95 \%\) level of confidence that the new medicine produces the effect desired.


9.8: Discussion Questions
How do funding requirements differ in each stage of a company’s lifecycle? Why?
2 .What are the advantages and disadvantages of different sources of financing?
3 .How does collateral factor into your financing options?
4 .What challenges do nonprofits face in securing financing?
5 .What are some potential benefits of a crowdfunding campaign aside from the funding itself? What are some potential drawbacks?
6 .What are some of the sacrifices you could see yourself having to make to bootstrap your potential business?
7 .How would you work around having limited access to capital in starting your own business? What difficulties can you see in trying to do so?
8 .Which external parties would be interested in obtaining a firm’s financial statements?
9 .Why would investors require access to financial statements?
10 .How do companies use financial statements in communicating with potential investors?
11 .How can an entrepreneur use preliminary financial information in planning a new business?


21.2.19: Chapter 19
Unfolding Case Study
1.
c.
The Mini Nutritional Assessment is a nutritional screening tool for older clients.
2.
a.
The client will need education regarding foods rich in iron to consume at home.
3.
b.
Using Maslow’s hierarchy of needs framework, understanding the physical environment should be the first consideration. It is essential to understand if the client will be cooking or eating prepared meals before considering other options.
4.
d.
Eating smaller portions more frequently is a strategy that can overcome decreased food intake that occurs due to getting full quickly. Although hypertension and diabetes are not risk factors for worsening anemia, decreased taste is a physiological consequence of the change in density of taste buds that occurs with aging.
5.
a.
A holistic approach to client teaching includes spiritual influences, emotional influences, physical influences, and social determinants of health.
6.
b.
The client’s cognitive functioning will determine if a caregiver needs to be present for the educational session.
7.
c.
Ms. Foster is choosing food options and making plans for meals. She will be in the action stage of change once she begins modifying her diet.
8.
a.
The nurse should recognize that change is a cyclic process and that successes and failures are expected.
Review Questions
1.
a.
Vitamin K is a fat-soluble vitamin necessary for the synthesis and activation of coagulation factors II (prothrombin), VII, IX, and X.
2.
c.
Folate deficiency in pregnancy can lead to macrocytic anemia in mothers and neural tube defects in fetuses.
3.
b.
Long-term alcohol use can cause the malabsorption of Vitamin B
12
.
4.
c.
These are all possible signs of scurvy, or vitamin C deficiency.
5.
c.
Iron is better absorbed when taken with vitamin C, found in orange juice, and on an empty stomach.
6.
a.
An older adult with anemia is likely to experience hypotension and/or lightheadedness, which is a significant risk for falls.
7.
a.
Animal meat, fish, eggs, and dairy are sources of vitamin B
12
.
8.
a.
Health literacy is the only modifiable psychosocial factor listed.
9.
a.
Providing written material before determining the client’s preferred learning style, without assessing their preference and without giving time to ask questions, is not consistent with holistic nursing care.
10.
a.
The nurse should consider that a change in behavior typically takes 6 months to achieve.
Unfolding Case Study
1.
c.
Often, a client with age-related degenerative disorders such as osteoarthritis will have limited ROM and increased pain upon ROM of the joints. This simple screening provides valuable information for the diagnosis of osteoarthritis as well as the treatment parameters for the client.
2.
a.
The provider would start with an x-ray of both knees to evaluate for any soft tissue changes and signs of arthritis.
3.
a.
Leafy green vegetables have anti-inflammatory properties. Because the client has osteoarthritis with edema in both knees, these nutrients should be included in the diet.
4.
b.
Nuts and seeds have anti-inflammatory properties that can be beneficial for clients with arthritis.
Review Questions
1.
a.
Adequate vitamin D intake is necessary for strong bones and the prevention and treatment of osteoporosis.
2.
d.
Good skin turgor indicates a normal moisture level in the skin. Tenting, when the skin remains stretched out like a tent when pinched, is a sign of decreased moisture in the skin.
3.
a.
Protein aids in the repair of muscles and tissue.
4.
d.
Bromelain is a group of enzymes found in pineapples and is believed to reduce swelling and pain.
5.
c.
The pressure ulcer changed from a stage 2 ulcer to a stage 1 ulcer, which indicates improvement. Stage 1 pressure injuries are reddened, but not open.
6.
b.
Foods rich in calcium and vitamin D, such as dairy products, are important for bone health and the treatment of osteoporosis.
7.
b.
Bread and pasta contain gluten and would not be included in a gluten-free diet.
8.
c.
Clubbing of nails, when nails look wider or swollen, is often seen with chronic obstructive pulmonary disease (COPD).
9.
b.
The client understands hydration, like drinking water throughout the day, promotes skin integrity and wound healing.
10.
d.
Biotin is found in foods such as liver and cooked egg yolks, and it is important in the promotion of hair and nail health.