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.

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.

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

Find the degree measure of the included angle, for each pair of vectors. Round your answers to the nearest tenth.
and .








What does the sign of the dot product tell us about the included angle?
Find all values of so that is orthogonal to . List your answers in increasing order.

Answer: .

Find the value of for which the vector is parallel to the vector . What is the measure of the included angle, ? Find the measure of the included angle using Theorem ex:anglebetweenvectors. Do the two results agree?


Prove that if is a unit vector, then .
Prove that if and are unit vectors, then . In what cases are the extreme values of 1 and attained?
Imagine a clock with hands represented by vectors and , as shown below. At what whole hour will attain its maximum value? At what whole hour will be as small as possible?



Let be a circle of radius . Suppose and are the endpoints of a diameter of , and is a point on distinct from and . Show that vectors and are orthogonal.
Assign coordinates to points , and , express vectors and in component form, then find the dot product of and .

A rhombus is a quadrilateral with four congruent sides. Use vectors to prove that a parallelogram is a rhombus if and only if its diagonals are perpendicular.
See section on vector subtraction in Module VEC-M-0030.

Photo Credits

The following images are courtesy of Wikimedia Commons

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