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: