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### Course: Precalculus > Unit 6

Lesson 7: Vector components from magnitude and direction- Vector components from magnitude & direction
- Vector components from magnitude & direction
- Vector components from magnitude & direction: word problem
- Vector components from magnitude & direction (advanced)
- Converting between vector components and magnitude & direction review

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# Converting between vector components and magnitude & direction review

Review how to find a vector's magnitude and direction from its components and vice versa.

## Cheat sheet

### Vector magnitude from components

The magnitude of $(a,b)$ is $||(a,b)||=\sqrt{{a}^{2}+{b}^{2}}$ .

### Vector direction from components

The direction angle of $(a,b)$ is $\theta ={\mathrm{tan}}^{-1}\left({\displaystyle \frac{b}{a}}\right)$ plus a correction based on the quadrant, according to this table:

Quadrant | How to adjust |
---|---|

Q1 | |

Q2 | |

Q3 | |

Q4 |

### Vector components from magnitude & direction

The components of a vector with magnitude $||\overrightarrow{u}||$ and direction $\theta $ are $(||\overrightarrow{u}||\mathrm{cos}(\theta ),||\overrightarrow{u}||\mathrm{sin}(\theta ))$ .

## What are vector magnitude and direction?

We are used to describing vectors in $(3,4)$ . We can plot vectors in the coordinate plane by drawing a directed line segment from the origin to the point that corresponds to the vector's components:

**component form**. For example,Considered graphically, there's another way to uniquely describe vectors — their ${\text{magnitude}}$ and ${\text{direction}}$ :

The ${\text{magnitude}}$ of a vector gives the length of the line segment, while the ${\text{direction}}$ gives the angle the line forms with the positive $x$ -axis.

The magnitude of vector $\overrightarrow{v}$ is usually written as $||\overrightarrow{v}||$ .

*Want to learn more about vector magnitude? Check out this video.*

*Want to learn more about vector direction? Check out this video.*

## Practice set 1: Magnitude from components

To find the magnitude of a vector from its components, we take the square root of the sum of the components' squares (this is a direct result of the Pythagorean theorem):

For example, the magnitude of $(3,4)$ is $\sqrt{{3}^{2}+{4}^{2}}=\sqrt{25}=5$ .

*Want to try more problems like this? Check out this exercise.*

## Practice set 2: Direction from components

To find the direction of a vector from its components, we take the inverse tangent of the ratio of the components:

This results from using trigonometry in the right triangle formed by the vector and the $x$ -axis.

### Example 1: Quadrant $\text{I}$

Let's find the direction of $(3,4)$ :

### Example 2: Quadrant $\text{IV}$

Let's find the direction of $(3,-4)$ :

The calculator returned a negative angle, but it's common to use positive values for the direction of a vector, so we must add ${360}^{\circ}$ :

### Example 3: Quadrant $\text{II}$

Let's find the direction of $(-3,4)$ . First, notice that $(-3,4)$ is in Quadrant $\text{II}$ .

*Want to try more problems like this? Check out this exercise.*

## Practice set 3: Components from magnitude and direction

To find the components of a vector from its magnitude and direction, we multiply the magnitude by the sine or cosine of the angle:

This results from using trigonometry in the right triangle formed by the vector and the $x$ -axis.

For example, this is the component form of the vector with magnitude ${2}$ and angle ${{30}^{\circ}}$ :

*Want to try more problems like this? Check out this exercise.*

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