Vector Magnitude and Direction Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Vector Magnitude and Direction.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The magnitude $\|\mathbf{v}\|$ is a vector's length; the direction is the angle it makes with a reference axis.

Magnitude is how long the arrow is—like measuring the length of a stick. Direction is which way it points. A unit vector is a 'pure direction' with length 1, like a compass needle. To get the unit vector, shrink or stretch the vector until its length is exactly 1 while keeping it pointed the same way.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Magnitude is the arrow's length from $\sqrt{v_1^2+v_2^2}$ ; direction is the angle it points.

Common stuck point: The procedure for vector magnitude and direction is the easy part; the trap is adding components for length. Asking "Am I asked how long the arrow is or which way it points, rather than how to combine arrows?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I asked how long the arrow is or which way it points, rather than how to combine arrows?

Worked Examples

Example 1

easy

Find the magnitude of

\mathbf{v} = \langle 3, 4 \rangle

Find the magnitude |v|

Answer

5

First step

Step 1:

\|\mathbf{v}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25}

Full solution

2
Step 2: $= 5$ .
3
Check: This is a 3-4-5 right triangle ✓

The magnitude (length) of a vector is found using the Pythagorean theorem:

\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2}

. This extends naturally to higher dimensions.

Example 2

medium

Find the unit vector in the direction of

\mathbf{v} = \langle 1, 2, 2 \rangle

Example 3

medium

Find the magnitude and direction angle of the vector v = <3, 4>.

Find |v| and direction angle θ

Example 4

medium

Find a unit vector in the direction of

\langle -1, 2, 2 \rangle

Example 5

medium

Write

\mathbf{v}

in component form if

\|\mathbf{v}\| = 10

and its direction angle is

60°

Write v in component form (|v|=10, θ=60°)

Example 6

hard

Find values of

k

so that

\langle k, k+1 \rangle

has magnitude

5

Example 7

hard

Show that

\|\mathbf{u} + \mathbf{v}\| \le \|\mathbf{u}\| + \|\mathbf{v}\|

for

\mathbf{u} = \langle 3, 0 \rangle

and

\mathbf{v} = \langle 0, 4 \rangle

Illustrate ||u+v|| ≤ ||u||+||v|| for u=⟨3,0⟩ and v=⟨0,4⟩

Example 8

hard

Decompose

\mathbf{v} = \langle 4, 3 \rangle

as a sum of a vector along

\langle 1, 0 \rangle

and a vector along

\langle 0, 1 \rangle

, and report each magnitude.

Example 9

challenge

Vectors

\mathbf{u}

and

\mathbf{v}

are perpendicular,

\|\mathbf{u}\| = 7

\|\mathbf{v}\| = 24

. Find

\|\mathbf{u} + \mathbf{v}\|

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Find

\|\langle -5, 12 \rangle\|

Find the magnitude of ⟨-5, 12⟩

Example 2

hard

Find the direction angle

\theta

\mathbf{v} = \langle -1, \sqrt{3} \rangle

measured from the positive

x

-axis.

Find direction angle θ of v

Example 3

easy

Find the magnitude of

\langle 3, 4 \rangle

Example 4

easy

Find the magnitude of

\langle 6, 8 \rangle

Find the magnitude of ⟨6, 8⟩

Example 5

easy

Find the magnitude of

\langle 0, 7 \rangle

Example 6

easy

Find the magnitude of

\langle -5, 0 \rangle

Example 7

easy

What is the magnitude of a unit vector?

Example 8

easy

Find the magnitude of

\langle 1, 1 \rangle

Example 9

easy

Can the zero vector

\langle 0, 0 \rangle

have a direction?

Example 10

easy

Find the magnitude of

\langle 8, 15 \rangle

Find the magnitude of ⟨8, 15⟩

Example 11

medium

Find the unit vector in the direction of

\langle 3, 4 \rangle

Example 12

medium

Find the direction angle of

\langle 1, 1 \rangle

from the positive

x

-axis.

Find the direction angle of ⟨1, 1⟩

Example 13

medium

Find the direction angle of

\langle -1, 1 \rangle

Find the direction angle of ⟨-1, 1⟩

Example 14

medium

A vector has magnitude 10 at

30^\circ

. Find its components.

Find the x- and y-components of v

Example 15

medium

Find the magnitude of

2\langle 3, 4 \rangle

Example 16

medium

Find the distance between points

(1, 2)

and

(4, 6)

using a vector.

Example 17

medium

\langle 0.6, 0.8 \rangle

a unit vector?

Example 18

medium

Find the magnitude of

\langle -3, -4 \rangle

Example 19

medium

Find the unit vector in the direction of

\langle 0, -2 \rangle

Example 20

challenge

Find a unit vector perpendicular to

\langle 3, 4 \rangle

Example 21

challenge

A vector has magnitude 13 and

x

-component 5 (first quadrant). Find its

y

-component.

Example 22

challenge

Two vectors

\langle 3, 0 \rangle

and

\langle 0, 4 \rangle

add to a resultant. Find the resultant's magnitude and direction angle.

Find the resultant magnitude and direction

Example 23

easy

Find

\|\langle 8, 15 \rangle\|

Find ||⟨8, 15⟩||

Example 24

easy

Find

\|\langle 2, 2, 1 \rangle\|

Example 25

easy

Find the direction angle of

\langle 1, 1 \rangle

Find the direction angle of ⟨1, 1⟩

Example 26

easy

Find the unit vector in the direction of

\langle 6, 8 \rangle

Example 27

medium

Find

\|\langle -3, 5, -4 \rangle\|

Example 28

medium

Find the direction angle of

\langle -3, 3 \rangle

measured from the positive

x

-axis.

Find the direction angle of ⟨-3, 3⟩

Example 29

medium

Find the direction angle of

\langle 0, -4 \rangle

Find the direction angle of ⟨0, -4⟩

Example 30

medium

Find the magnitude of

\langle 3, -4 \rangle + \langle 0, 1 \rangle

Example 31

medium

A plane flies at

300

km/h in direction

30°

north of east. Write its velocity in component form.

Write the plane's velocity in component form

Example 32

hard

Find the direction angle of

\langle -2, -2\sqrt{3} \rangle

measured from the positive

x

-axis.

Example 33

hard

Find a vector of magnitude

13

in the direction of

\langle 5, 12 \rangle

Example 34

hard

Two forces of magnitude

10

N act at

0°

and

120°

. Find the magnitude of the resultant.

Find the magnitude of the resultant of two 10N forces at 0° and 120°

Example 35

hard

A vector

\mathbf{v}

has magnitude

8

and points in the same direction as

\langle 1, -1, 2 \rangle

. Write

\mathbf{v}

in component form.

Example 36

hard

Find

\|\mathbf{u}\|

\mathbf{u} - 2\langle 1, -3 \rangle = \langle 5, 1 \rangle

Example 37

challenge

Find the direction angle (from positive

x

-axis,

0° \le \theta < 360°

) of

\mathbf{v} = \langle 3, -4 \rangle

Find the direction angle (0°≤θ<360°) of v

← Back to Vector Magnitude and Direction Formula Explained Practice Mode

Related Concepts

Vector Addition, Subtraction, and Scalar Multiplication Simplifying Radicals Dot Product Cross Product

Background Knowledge

These ideas may be useful before you work through the harder examples.

vector operationssimplifying radicals