Vectors Examples: 47 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Vectors.

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

Concept Recap

Mathematical quantities that possess both a magnitude (size) and a direction, represented graphically as arrows.

An arrow pointing somewhere with a certain length—the length is 'how much,' the direction is 'which 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: Vectors starts by naming what changes, over what time interval, and whether direction matters.

Common stuck point: Students often know a formula related to vectors but skip the recognition step: Am I describing motion over time with position, distance, direction, speed, velocity, or acceleration clearly separated? That leads to a correct-looking substitution attached to the wrong physical model.

Sense of Study hint: Ask: Am I describing motion over time with position, distance, direction, speed, velocity, or acceleration clearly separated?

Common Mistakes to Watch For

Before you work through the examples, skim the mistake guide so you know which shortcuts and sign errors to avoid.

Units and Signs Mistakes

Worked Examples

Example 1

easy

Find the magnitude and direction of the vector with components

v_x = 3

and

v_y = 4

Resolve into components; find the resultant magnitude and angle θ

Answer

∣ \vec{v} ∣ = 5, θ \approx 53.1 °

First step

Use the Pythagorean theorem for the magnitude of the vector.

Full solution

2
Magnitude: $∣ \vec{v} ∣ = \sqrt{v_{x}^{2} + v_{y}^{2}} = \sqrt{9 + 16} = \sqrt{25} = 5$
3
Direction: $θ = \tan^{- 1} (\frac{v_{y}}{v_{x}}) = \tan^{- 1} (\frac{4}{3}) \approx 53.1 °$

Any vector can be described by its magnitude and direction or by its components. The Pythagorean theorem gives the magnitude, and the inverse tangent gives the angle.

Example 2

medium

Add the vectors:

\vec{A} = 5 \text{ m}

0°

(east) and

\vec{B} = 8 \text{ m}

90°

(north). Find the resultant.

Add vectors head-to-tail; find the magnitude and direction of the resultant

Example 3

medium

Add the vectors

\vec{A} = (3, 4)

and

\vec{B} = (-1, 2)

component-wise, then report the resultant's magnitude.

Add vectors A=(3,4) and B=(−1,2) head-to-tail; find the resultant magnitude

Example 4

medium

Two displacements

\vec{d}_1 = 10 \text{ m}

east and

\vec{d}_2 = 10 \text{ m at } 60°

north of east act on an object. Find the magnitude of the resultant.

Find the magnitude of the resultant of 10 m east and 10 m at 60° north of east

Example 5

medium

A swimmer crosses a river by aiming at

90°

to the current. Her swim speed relative to water is

1.2 \text{ m/s}

and the current is

0.5 \text{ m/s}

. Find her speed relative to ground.

Find the swimmer's speed relative to the ground

Example 6

medium

Find the angle that

\vec{v} = (5, 12)

makes with the

+x

-axis.

Find the angle that v = (5, 12) makes with the +x axis

Example 7

hard

Two forces of equal magnitude

F

act at a point with a

120°

angle between them. Find the magnitude of their resultant in terms of

F

Find the resultant magnitude when two equal forces F act at 120° between them

Example 8

hard

An airplane needs to fly due north at

300 \text{ km/h}

ground speed. A wind blows at

50 \text{ km/h}

due east. What heading should the pilot aim?

Example 9

hard

A boat aims at

30°

west of north relative to the water at

4 \text{ m/s}

in a river flowing

2 \text{ m/s}

east. Find the boat's speed relative to the ground.

Practice Problems

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

Example 1

medium

A force of

50 \text{ N}

acts at

60°

above the horizontal. Find the horizontal and vertical components.

Find the horizontal and vertical components of the 50 N force

Example 2

medium

Two forces act on an object:

\vec{F}_1 = 8 \text{ N}

east and

\vec{F}_2 = 6 \text{ N}

north. Find the magnitude and direction of the resultant force.

Find the magnitude and direction of the resultant force

Example 3

easy

Two perpendicular forces

3

N and

4

N act on a point. Find the resultant magnitude.

Find the resultant magnitude of the two perpendicular forces

Example 4

easy

50

km/h a vector or a scalar?

Example 5

easy

A vector points

6

units east. What is its magnitude?

Example 6

easy

Resolve a

10

N force at

0°

(along

x

) into

x

and

y

components.

Example 7

easy

Two vectors

5

east and

5

west add to what?

Example 8

easy

A displacement vector is '

7

m'. What must be added to make it a proper vector?

Example 9

easy

Which component of a vector at angle

\theta

uses cosine: the one along the angle's axis or perpendicular?

Example 10

easy

Add two collinear vectors:

8

N east and

3

N east.

Example 11

medium

Add vectors

6

N east and

8

N north. Find magnitude and confirm the triple.

Add 6 N east and 8 N north; find the resultant magnitude

Example 12

medium

A vector has magnitude

10

30°

above horizontal. Find its horizontal and vertical components (

\cos30°\approx0.866

\sin30°=0.5

Find the horizontal and vertical components of v

Example 13

medium

Find the magnitude of the resultant of

9

N east and

12

N north.

Find the resultant magnitude of 9 N east and 12 N north

Example 14

medium

Subtract vectors:

\vec A=10

east,

\vec B=4

east. Find

\vec A-\vec B

Example 15

medium

A force has components

F_x=5

N and

F_y=12

N. Find its magnitude.

Find the magnitude of the force from its components Fx

Example 16

medium

Two equal vectors of magnitude

5

act at

90°

to each other. Find the resultant magnitude.

Find the resultant magnitude of two equal 5 N forces at 90° to each other

Example 17

challenge

Three vectors:

3

east,

4

north,

3

west. Find the resultant magnitude.

Find the resultant magnitude of 3 east + 4 north + 3 west

Example 18

challenge

Vector

\vec A

7

east;

\vec B

24

north. Find the angle of

\vec A+\vec B

above east (use

\tan^{-1}

Find the angle that A + B makes above east

Example 19

challenge

A boat is pushed

8

N east by motor and

6

N south by current, plus

6

N north by a tug. Net force magnitude and direction?

Find the net force on the boat from motor 8N east, current 6N south, and tug 6N north

Example 20

medium

A vector has components

F_x=8

F_y=15

N. Find its magnitude.

Find the magnitude of the force from components Fx

Example 21

medium

Add

4

N east and

4

N east and

3

N west. Find the resultant.

Example 22

medium

Resolve a

26

N force with

\cos\theta=5/13

\sin\theta=12/13

into components.

Resolve the 26 N force into its horizontal and vertical components

Example 23

easy

A vector has components

v_x = -6

and

v_y = 8

. Find its magnitude.

Find the magnitude of vector v with components vx

Example 24

easy

Find the components of a

20 \text{ N}

force directed at

30°

above the horizontal.

Find the components of the 20 N force directed at 30° above horizontal

Example 25

easy

A vector has

v_x = 0

and

v_y = -7

. State its magnitude and direction.

State the magnitude and direction of the vector with vx

Example 26

medium

A hiker walks

4 \text{ km}

north then

3 \text{ km}

east. Find the magnitude and direction of his displacement from the start.

Find the hiker's displacement magnitude and direction from start

Example 27

medium

Subtract:

\vec{A} - \vec{B}

where

\vec{A} = (7, 2)

and

\vec{B} = (3, 5)

Example 28

medium

15 \text{ N}

force points at

120°

measured counter-clockwise from the

+x

-axis. Find its components.

Find the components of the 15 N force at 120° from the +x axis

Example 29

medium

\vec{v} = (6, -8)

, find a unit vector in the same direction.

Example 30

medium

Three coplanar forces act on a point:

\vec{F}_1 = (4, 0) \text{ N}

\vec{F}_2 = (0, 3) \text{ N}

\vec{F}_3 = (-4, -3) \text{ N}

. Find the net force.

Example 31

medium

A car drives

5 \text{ km}

east, then

5 \text{ km}

south, then

5 \text{ km}

west. What is its displacement?

Example 32

medium

Multiply the vector

(2, -3)

by the scalar

-4

Example 33

hard

100 \text{ N}

box is pulled up a ramp by a rope at

25°

above the ramp surface. Find the component of the rope's force perpendicular to the ramp.

Example 34

hard

A plane is heading

30°

east of north at

200 \text{ km/h}

relative to the air. A wind blows due east at

40 \text{ km/h}

. Find the magnitude of the ground velocity.

Example 35

hard

Find the dot product

\vec{A} \cdot \vec{B}

for

\vec{A} = (3, 2)

and

\vec{B} = (-4, 6)

Example 36

hard

50 \text{ N}

vector lies in the

xy

-plane at

45°

from

+x

. A second vector

\vec{B}

added to it gives the resultant

(0, 0)

. Find

\vec{B}

Example 37

hard

Vectors

\vec{A}

and

\vec{B}

have magnitudes

6

and

8

and are at right angles. Find the magnitude of

\vec{A} - \vec{B}

Example 38

challenge

Three forces in equilibrium have magnitudes

5 \text{ N}

12 \text{ N}

, and

13 \text{ N}

. What is the angle between the

5 \text{ N}

and

12 \text{ N}

forces?

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Related Concepts

Displacement Velocity Force