# Vector magnitude and direction review

Review your knowledge of vector magnitude and direction, and use them to solve problems.

Magnitude of $(a,b)$ | |

$\mid\mid\!\!\!(a,b)\!\!\mid\mid=\sqrt{a^2+b^2}$ | |

Direction of $(a,b)$ | |

$\theta=\tan^{-1}\left(\dfrac{b}{a}\right)$ | |

Components from magnitude $\mid\mid\!\!\!\vec u\!\!\mid\mid$ and direction $\theta$ | |

$\bigg(\mid\mid\!\!\!\vec u\!\!\mid\mid\cos(\theta),\mid\mid\!\!\!\vec u\!\!\mid\mid\sin(\theta)\bigg)$ |

## What are vector magnitude and direction?

We are used to describing vectors in

**component form**. For example, $(3,4)$. 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:Considered graphically, there's another way to uniquely describe vectors — their $\blueD{\text{magnitude}}$ and $\greenD{\text{direction}}$:

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

The magnitude of vector $\vec v$ is usually written as $||\vec v||$.

*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 $\text{II}$.

$-53^\circ$ is in Quadrant $\text{IV}$, not $\text{II}$. We must add $180^\circ$ to obtain the opposite angle:

*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 $\blueD 2$ and angle $\greenD{30^\circ}$:

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