In this explainer, we will learn how to find direction angles and direction cosines for a given vector in space.
We know, in three-dimensional coordinate space, we have the -, -, and -axes. These are perpendicular to one another as seen in the diagram below. The unit vectors , , and act in the -, -, and -directions respectively.
Definition: The Direction Angles
Given a vector , the angles that this vector makes with the -, -, and -axes, respectively, are , , and . These are known as the direction angles and are written .
These direction angles lead us to a definition for the direction cosines. We know, in right-angled trigonometry, the cosine of any angle is equal to the length of the side adjacent to the angle divided by the length of the hypotenuse:
Definition: The Direction Cosines
The direction cosines are the cosines of the three direction angles, , , and :
, , and are the -, -, and -components of vector , and is the magnitude or norm of this vector, where
We can rearrange the three equations such that
In our first example, we will demonstrate how to find a vector given its direction angles and magnitude.
Example 1: Finding a Vector given Its Norm and Direction Angles
Find vector whose norm is 41 and whose direction angles are .
Answer
The norm of a vector is its magnitude or length, and the direction angles , , and are the angles between the vector and the -, -, and -axes respectively.
Using the formulae for the direction cosines, we know that, for a vector ,
Multiplying each of these equations by , the components of are as follows:
Therefore, .
Before looking at our next example, we will consider a formula that links the direction cosines.
Consider the direction cosines as follows:
Squaring both sides of these three equations gives us
Adding these three equations, we get
We also know that
So,
This means that the right-hand side of the equation is equal to 1.
Therefore,
Formula: Property of Direction Cosines for a Three-Dimensional Vector
If , , and are the three direction angles and , , and are their corresponding direction cosines, then
Example 2: Finding the Third Direction Angle of a Vector
Suppose that , , and are the direction angles of a vector. Which of the following, to the nearest hundredth, is ?
Answer
In order to answer this question, we will use the fact that if the three direction angles of a vector are , , and , then
If we let , , and , we have,
Taking the square root of both sides of our equation,
Taking the inverse cosine of both sides,
So, the value of , to the nearest hundredth, is .
In our next example, we will demonstrate how to calculate the direction cosines of a vector.
Example 3: Finding the Direction Cosines of a Vector
Find the direction cosines of the vector .
Answer
Given any vector with components , , and , the direction cosines are where
Substituting , , and , we have
Therefore,
The direction cosines of the vector are .
Example 4: Finding the Direction Angles of a Vector
Find the direction angles of the vector .
Answer
Given any vector with components , , and , the direction cosines are where
Substituting , , and , we have
Therefore, which simplify to
Taking the inverse cosine of both sides of these three equations,
The direction angles of the vector are .
Example 5: Finding the Direction Angles of a Vector
Find the measure of the direction angles of the vector , represented by the given figure, corrected to one decimal place.
Answer
We begin by writing the vector in terms of its three components, where we will define 1 unit to be 1 cm.
In the -direction, we travel 8 cm, so the -component is 8. In the -direction, we travel 19 cm, so the -component is 19. In the -direction, we travel 9 cm, so the -component is 9:
The magnitude of the vector is where , , and are the -, -, and -components of vector :
Given any vector with components , , and and direction angles , , and ,
So,
Therefore, the direction angles of the vector are , , .
We will finish this explainer by recapping some of the key points.
Key Points
- The direction angles , , and are the angles between a vector and the -, -, and -axes respectively.
- The direction cosines are the cosines of the three direction angles, , , and , such that
- This means that , , and are as follows:
- The following formula links the three direction cosines:
