We state and prove the cosine formula for the dot product of two vectors, and show that two vectors are orthogonal if and only if their dot product is zero.

### VEC-0060: Dot Product and the Angle Between Vectors

Given two vectors and , let be the angle between them such that . We will refer to
as the *included angle*.

The following theorem establishes a relationship between the dot product and the included angle.

- Proof
- Consider the triangle formed by , and .
By the Law of Cosines we have:

By Theorem th:dotproductpropertiesitem:norm of VEC-0050

By Theorem th:dotproductpropertiesitem:distributive-again of VEC-0050

By Theorem th:dotproductpropert iesitem:distributive of VEC-0050

By Theorem th:dotproductpropert iesitem:commutative of VEC-0050

#### Orthogonal Vectors

Two non-zero vectors are said to be *perpendicular* or *orthogonal* if the included angle
is a right angle.

- Proof
- Recall that to prove an “if and only if” statement, we must prove two
statements. In this particular case, we will need to show the following
- If , then and are orthogonal.
- If and are orthogonal, then .

We leave the details of the proof to the reader.

### Practice Problems

Answer:

Answer:

### Photo Credits

The following images are courtesy of Wikimedia Commons

Hannes Grobe, Wall clock manufactured by Telefonbau & Normalzeit. CC-BY 3.0