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 ∥v∥\|\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 v12+v22\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 v=⟨3,4⟩\mathbf{v} = \langle 3, 4 \rangle.

Answer

55

First step

1
Step 1: ∥v∥=32+42=9+16=25\|\mathbf{v}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25}.

Full solution

  1. 2
    Step 2: =5= 5.
  2. 3
    Check: This is a 3-4-5 right triangle ✓
The magnitude (length) of a vector is found using the Pythagorean theorem: ∥v∥=v12+v22\|\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 v=⟨1,2,2⟩\mathbf{v} = \langle 1, 2, 2 \rangle.

Example 3

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

Example 4

medium
Find a unit vector in the direction of ⟨−1,2,2⟩\langle -1, 2, 2 \rangle.

Example 5

medium
Write v\mathbf{v} in component form if ∥v∥=10\|\mathbf{v}\| = 10 and its direction angle is 60°60°.

Example 6

hard
Find values of kk so that ⟨k,k+1⟩\langle k, k+1 \rangle has magnitude 55.

Example 7

hard
Show that ∥u+v∥≤∥u∥+∥v∥\|\mathbf{u} + \mathbf{v}\| \le \|\mathbf{u}\| + \|\mathbf{v}\| for u=⟨3,0⟩\mathbf{u} = \langle 3, 0 \rangle and v=⟨0,4⟩\mathbf{v} = \langle 0, 4 \rangle.

Example 8

hard
Decompose v=⟨4,3⟩\mathbf{v} = \langle 4, 3 \rangle as a sum of a vector along ⟨1,0⟩\langle 1, 0 \rangle and a vector along ⟨0,1⟩\langle 0, 1 \rangle, and report each magnitude.

Example 9

challenge
Vectors u\mathbf{u} and v\mathbf{v} are perpendicular, ∥u∥=7\|\mathbf{u}\| = 7, ∥v∥=24\|\mathbf{v}\| = 24. Find ∥u+v∥\|\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 ∥⟨−5,12⟩∥\|\langle -5, 12 \rangle\|.

Example 2

hard
Find the direction angle θ\theta of v=⟨−1,3⟩\mathbf{v} = \langle -1, \sqrt{3} \rangle measured from the positive xx-axis.

Example 3

easy
Find the magnitude of ⟨3,4⟩\langle 3, 4 \rangle.

Example 4

easy
Find the magnitude of ⟨6,8⟩\langle 6, 8 \rangle.

Example 5

easy
Find the magnitude of ⟨0,7⟩\langle 0, 7 \rangle.

Example 6

easy
Find the magnitude of ⟨−5,0⟩\langle -5, 0 \rangle.

Example 7

easy
What is the magnitude of a unit vector?

Example 8

easy
Find the magnitude of ⟨1,1⟩\langle 1, 1 \rangle.

Example 9

easy
Can the zero vector ⟨0,0⟩\langle 0, 0 \rangle have a direction?

Example 10

easy
Find the magnitude of ⟨8,15⟩\langle 8, 15 \rangle.

Example 11

medium
Find the unit vector in the direction of ⟨3,4⟩\langle 3, 4 \rangle.

Example 12

medium
Find the direction angle of ⟨1,1⟩\langle 1, 1 \rangle from the positive xx-axis.

Example 13

medium
Find the direction angle of ⟨−1,1⟩\langle -1, 1 \rangle.

Example 14

medium
A vector has magnitude 10 at 30∘30^\circ. Find its components.

Example 15

medium
Find the magnitude of 2⟨3,4⟩2\langle 3, 4 \rangle.

Example 16

medium
Find the distance between points (1,2)(1, 2) and (4,6)(4, 6) using a vector.

Example 17

medium
Is ⟨0.6,0.8⟩\langle 0.6, 0.8 \rangle a unit vector?

Example 18

medium
Find the magnitude of ⟨−3,−4⟩\langle -3, -4 \rangle.

Example 19

medium
Find the unit vector in the direction of ⟨0,−2⟩\langle 0, -2 \rangle.

Example 20

challenge
Find a unit vector perpendicular to ⟨3,4⟩\langle 3, 4 \rangle.

Example 21

challenge
A vector has magnitude 13 and xx-component 5 (first quadrant). Find its yy-component.

Example 22

challenge
Two vectors ⟨3,0⟩\langle 3, 0 \rangle and ⟨0,4⟩\langle 0, 4 \rangle add to a resultant. Find the resultant's magnitude and direction angle.

Example 23

easy
Find ∥⟨8,15⟩∥\|\langle 8, 15 \rangle\|.

Example 24

easy
Find ∥⟨2,2,1⟩∥\|\langle 2, 2, 1 \rangle\|.

Example 25

easy
Find the direction angle of ⟨1,1⟩\langle 1, 1 \rangle.

Example 26

easy
Find the unit vector in the direction of ⟨6,8⟩\langle 6, 8 \rangle.

Example 27

medium
Find ∥⟨−3,5,−4⟩∥\|\langle -3, 5, -4 \rangle\|.

Example 28

medium
Find the direction angle of ⟨−3,3⟩\langle -3, 3 \rangle measured from the positive xx-axis.

Example 29

medium
Find the direction angle of ⟨0,−4⟩\langle 0, -4 \rangle.

Example 30

medium
Find the magnitude of ⟨3,−4⟩+⟨0,1⟩\langle 3, -4 \rangle + \langle 0, 1 \rangle.

Example 31

medium
A plane flies at 300300 km/h in direction 30°30° north of east. Write its velocity in component form.

Example 32

hard
Find the direction angle of ⟨−2,−23⟩\langle -2, -2\sqrt{3} \rangle measured from the positive xx-axis.

Example 33

hard
Find a vector of magnitude 1313 in the direction of ⟨5,12⟩\langle 5, 12 \rangle.

Example 34

hard
Two forces of magnitude 1010 N act at 0°0° and 120°120°. Find the magnitude of the resultant.

Example 35

hard
A vector v\mathbf{v} has magnitude 88 and points in the same direction as ⟨1,−1,2⟩\langle 1, -1, 2 \rangle. Write v\mathbf{v} in component form.

Example 36

hard
Find ∥u∥\|\mathbf{u}\| if u−2⟨1,−3⟩=⟨5,1⟩\mathbf{u} - 2\langle 1, -3 \rangle = \langle 5, 1 \rangle.

Example 37

challenge
Find the direction angle (from positive xx-axis, 0°≤θ<360°0° \le \theta < 360°) of v=⟨3,−4⟩\mathbf{v} = \langle 3, -4 \rangle.
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Background Knowledge

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

vector operationssimplifying radicals