Practice Vector Magnitude and Direction in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick 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.

Showing a random 20 of 50 problems.

Example 1

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

Example 2

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

Example 3

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

Example 4

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

Example 5

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}\|.

Example 6

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

Example 7

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

Example 8

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 9

easy
What is the magnitude of ⟨0,0,0⟩\langle 0, 0, 0 \rangle?

Example 10

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

Example 11

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.

Example 12

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

Example 13

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

Example 14

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

Example 15

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

Example 16

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

Example 17

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

Example 18

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

Example 19

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

Example 20

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