Section Summary: 13.3 - The dot product

1. Definitions

   If a and b , then the dot product of a and b is the number given by

   

   Two non-zero vectors a and b are called perpendicular or orthogonal if the angle between them is .

   The direction angles of a non-zero vector a are the angles , , and that the vector makes with the positive x-, y-, and z-axes. The cosines of these angles are called the direction cosines.

   The vector projection of b onto a is

   

   The scalar projection of b onto a is

   

   In particular, the unit vector created from a would be

   

   The vector with the same direction, but unit length.

2. Theorems

   If is the angle between vectors a and b, then

   

   Corollary:

   

   Hence, a and b are orthogonal (i.e. perpendicular)

3. Properties/Tricks/Hints/Etc.

   Properties of the dot product:

   

4. Summary

   The dot product is a way of combining two vectors to get a scalar (i.e., a number). It effectively measures the shadow that one vector casts on the other, and if the dot product is zero, the two vectors are orthogonal (i.e. perpendicular).