Dot Product and Angle

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 Dot Product and its Properties By Theorem th:dotproductpropertiesitem:distributive-again of Dot Product and its Properties By Theorem th:dotproductpropert iesitem:distributive of Dot Product and its Properties By Theorem th:dotproductpropert iesitem:commutative of Dot Product and its Properties

Orthogonal Vectors

We can use Theorem th:dotproductcosine to show that two non-zero orthogonal vectors of are simply perpendicular vectors (the included angle is ). To see this, suppose that for nonzero vectors . Then from Theorem th:dotproductcosine we have Since are nonzero vectors, we have , which implies . The converse also holds. If , then the dot product is clearly 0.

The reason we prefer the term “orthogonal” to “perpendicular” in this course is because is only one example of a vector space, and the dot product is only one example of a more general product, called an inner product. For vectors in a zero dot product happens to coincide with the geometric idea of perpendicularity, but there are many vector spaces that do not possess the visual geometry of . (Later in the text, you will encounter vector spaces whose vectors are polynomial functions!) In these more abstract settings, a zero inner product still signals a special relationship between vectors. The term orthogonal captures this relationship.

Practice Problems

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

Answer:

and

Answer:

and

Answer:

and

Answer:

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?

Answer:

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?

PIC

Answer:

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 Vector Arithmetic.
The points , and form a triangle in . Is it a right triangle?
Express each side of the triangle as a vector and use what you have learned in this section.

Photo Credits

The following images are courtesy of Wikimedia Commons

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