<s>[INST] Demonstrate that the two formulae for a scalar product are equivalent. In the figure below, three vectors are joined together to form a triangle.
The name of each vector is a single letter in boldface,
each vector is specified by three lengths in an $xyz$ coordinate system,
and vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ are separated by angle $\theta$.

Let $a$, $b$, and $c$ (with neither boldface nor subscripts) represent the 
magnitudes (lengths) of vectors $\boldsymbol{a}$, $\boldsymbol{b}$, and $\boldsymbol{c}$.
Note that the Pythagorean theorem is used.
\begin{eqnarray*}
  a  &=&  \sqrt{a_x^2 + a_y^2 + a_z^2}  \\
  b  &=&  \sqrt{b_x^2 + b_y^2 + b_z^2}  \\
  c  &=&  \sqrt{c_x^2 + c_y^2 + c_z^2}  
\end{eqnarray*}
The expression $\boldsymbol{a} \cdot \boldsymbol{b}$
signifies the $scalar\ product$ (a.k.a. the $dot\ product$) 
of vectors $\boldsymbol{a}$ and $\boldsymbol{b}$.
There are two common formulas for computing $\boldsymbol{a} \cdot \boldsymbol{b}$.
The first is the sum of the products of the $xyz$ components.
The second is the product of the magnitudes and the cosine of angle $\theta$.
\begin{eqnarray*}
  \boldsymbol{a} \cdot \boldsymbol{b} &=& a_x b_x + a_y b_y + a_z b_z \\
  \boldsymbol{a} \cdot \boldsymbol{b} &=& a b \cos \theta  
\end{eqnarray*}
It's not obvious that these two formulas for the scalar product 
are equivalent.  Therefore, demonstrate that they are --- 
demonstrate that
\begin{eqnarray*}
  a_x b_x + a_y b_y + a_z b_z &=& a b \cos \theta
\end{eqnarray*}
 [/INST] Vector $\boldsymbol{a}$ is the sum of vectors $\boldsymbol{b}$ and $\boldsymbol{c}$.
I.e., $\boldsymbol{a} = \boldsymbol{b} + \boldsymbol{c}$.  Solving for $\boldsymbol{c}$ we have
\begin{eqnarray*}
  \boldsymbol{c} &=& \boldsymbol{a} - \boldsymbol{b} \\
         &=& (a_x, a_y, a_z) - (b_x, b_y, b_z) \\
         &=& \big( (a_x-b_x),(a_y-b_y),(a_z-b_z) \big) \\
         &=& \big( c_x , c_y , c_z \big)
\end{eqnarray*}
We calculate magnitude $c$
using the $xyz$ components of vectors $\boldsymbol{a}$ and $\boldsymbol{b}$.
\begin{eqnarray*}
  c &=& \sqrt{c_x^2 + c_y^2 + c_z^2} \\
    &=& \sqrt{(a_x-b_x)^2 + (a_y-b_y)^2 + (a_z-b_z)^2}
\end{eqnarray*}
We start with the squared magnitude $c^2$, expand it, rearrange it, 
then reduce it with the squared magnitudes $a^2$ and $b^2$.
\begin{eqnarray*}
  c^2 &=& (a_x-b_x)^2 + (a_y-b_y)^2 + (a_z-b_z)^2 \\
      &=& (a_x^2-2a_xb_x+b_x^2) + (a_y^2-2a_yb_y+b_y^2) + (a_z^2-2a_zb_z+b_z^2) \\
      &=& (a_x^2+a_y^2+a_z^2) + (b_x^2+b_y^2+b_z^2) - 2(a_xb_x+a_yb_y+a_zb_z) \\
      &=& a^2 + b^2 - 2(a_xb_x+a_yb_y+a_zb_z)
\end{eqnarray*}
The Law of Cosines asserts that $c^2 = a^2 + b^2 - 2ab\cos\theta$.
We replace $c^2$ with the result from above, subtract $a^2 + b^2$ from both sides, 
divide both sides by $-2$, and we're done.
\begin{eqnarray*}
  \mathrm{Law\ of\ Cosines:\phantom{X}} c^2 &=& a^2 + b^2 - 2ab\cos\theta \\
  a^2 + b^2 - 2(a_xb_x+a_yb_y+a_zb_z) &=& a^2 + b^2 - 2ab\cos\theta \\
  -2(a_xb_x+a_yb_y+a_zb_z) &=& -2ab\cos\theta \\
  a_xb_x + a_yb_y + a_zb_z &=& ab\cos\theta
\end{eqnarray*}
QED
</s>