section_path: In linear algebra/Dot product, angle, and length
section_name: Dot product, angle, and length
paragraph_text: Dot product, angle, and length
  
  A vector can be pictured as an arrow. The vector's magnitude is its length, and its direction is the direction the arrow points. A vector in R 3 {\displaystyle \mathbb {R} ^{3}} can be represented by an ordered triple of real numbers. These numbers are called the components of the vector.
  
  The dot product of two vectors A = and B = is defined as:
  
  The magnitude of a vector A is denoted by || A ||. The dot product of a vector A = with itself is
  
  which gives
  
  the formula for the Euclidean length of the vector.
  
  Without reference to the components of the vectors, the dot product of two non-zero Euclidean vectors A and B is given by
  
  where θ is the angle between A and B.