## 17Calculus - Dot Product Application - Direction Cosines and Direction Angles

Using Vectors

Applications

### Partial Integrals

Double Integrals - 2Int

Triple Integrals - 3Int

Practice

### Articles

Direction angles are the angles between a given vector $$\vec{v}$$ and each coordinate axis (usually in three dimensions, so there are three of them). Basically, we use the equation for the angle between vectors to get the direction cosine equations and the direction angles. For example, to find the direction cosine and the direction angle between a vector $$\vec{v}$$ and the x-axis, we have
$$\displaystyle{ \cos(\alpha) = \frac{\vec{v} \cdot \hat{i}}{\norm{\vec{v}} \norm{\hat{i}}} }$$
Let's label the components of $$\vec{v}$$ as $$\vec{v} = v_1\hat{i} + v_2\hat{j} + v_3\hat{k}$$
Since $$\hat{i}$$ is the unit vector in the direction of the x-axis, we can write $$\hat{i} = 1\hat{i} + 0\hat{j} + 0\hat{k}$$ and $$\|\hat{i}\| = 1$$.
$$\displaystyle{ \cos(\alpha) = \frac{\vec{v} \cdot \hat{i}}{\|\vec{v}\| \|\hat{i}\|} = \frac{v_1}{\|\vec{v}\|} }$$
Similar results can be obtained for the other two angles. Most textbooks and mathematicians use special greek letters for these angles as listed below.

Angle Description Direction Cosine Direction Angle $$\alpha$$ is the angle between $$\vec{v}$$ and $$\hat{i}$$ $$\displaystyle{ \cos(\alpha) = \frac{v_1}{\|\vec{v}\|} }$$ $$\displaystyle{ \alpha = \arccos\left(\frac{v_1}{\|\vec{v}\|} \right) }$$ $$\beta$$ is the angle between $$\vec{v}$$ and $$\hat{j}$$ $$\displaystyle{ \cos(\beta) = \frac{v_2}{\|\vec{v}\|} }$$ $$\displaystyle{ \beta = \arccos\left(\frac{v_2}{\|\vec{v}\|} \right) }$$ $$\gamma$$ is the angle between $$\vec{v}$$ and $$\hat{k}$$ $$\displaystyle{ \cos(\gamma) = \frac{v_3}{\|\vec{v}\|} }$$ $$\displaystyle{ \gamma = \arccos\left(\frac{v_3}{\|\vec{v}\|} \right) }$$

Study Hint: Since you already need to know the equation for the angle between two vectors, just remember what the direction cosines and direction angles are. You can derive the equations in the above table from that information. Additionally, if you just memorize the equations, you may not remember what they represent or where they come from. What are you going to do when your instructor asks you to define they mean and where they come from?

Practice

Find the direction angle of $$3 \vhat{i} - 4\vhat{j}$$.

Problem Statement

Find the direction angle of $$3 \vhat{i} - 4\vhat{j}$$.

Solution

### 1809 video

video by Larson Calculus

Log in to rate this practice problem and to see it's current rating.

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.