Vector Operations Examples

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Vector Addition, Subtraction, and Scalar Multiplication.

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

Concept Recap

Vectors are added and subtracted component by component. Scalar multiplication multiplies each component of a vector by a number. If $\mathbf{u} = \langle u_1, u_2 \rangle$ and $\mathbf{v} = \langle v_1, v_2 \rangle$ , then $\mathbf{u} + \mathbf{v} = \langle u_1 + v_1, u_2 + v_2 \rangle$ and $k\mathbf{u} = \langle ku_1, ku_2 \rangle$ .

Vectors are arrows with direction and magnitude. Adding two vectors is like walking along the first arrow, then continuing along the second—you end up at the tip of the combined arrow (tip-to-tail method). Scalar multiplication stretches or shrinks the arrow: $2\mathbf{v}$ is twice as long in the same direction, while $-\mathbf{v}$ points the opposite 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 add and subtract component by component, and a scalar multiplies every component.

Common stuck point: The procedure for vector addition, subtraction, and scalar multiplication is the easy part; the trap is adding magnitudes instead of components. Asking "Am I adding/subtracting matching components, or multiplying one vector by a single number?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I adding/subtracting matching components, or multiplying one vector by a single number?

Worked Examples

Example 1

easy

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

and

\mathbf{v} = \langle 1, 4 \rangle

, find

2\mathbf{u} - \mathbf{v}

Answer

\langle 5, -6 \rangle

First step

Step 1:

2\mathbf{u} = \langle 6, -2 \rangle

Full solution

2
Step 2: $2\mathbf{u} - \mathbf{v} = \langle 6 - 1, -2 - 4 \rangle = \langle 5, -6 \rangle$ .
3
Check: Each component is $2u_i - v_i$ ✓

Scalar multiplication scales each component, then subtraction is done component-wise. Linear combinations of vectors like

a\mathbf{u} + b\mathbf{v}

are fundamental in linear algebra.

Example 2

medium

Find

\mathbf{u} - \mathbf{v}

where

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

and

\mathbf{v} = \langle 4, -3, 2 \rangle

Example 3

medium

Given u = <2, -1> and v = <3, 5>, find 2u - v and its magnitude.

Example 4

medium

Find scalars

s

and

t

so that

s\langle 1, 0 \rangle + t\langle 0, 1 \rangle = \langle 7, -4 \rangle

Example 5

medium

Show that

\mathbf{u} = \langle 4, 6 \rangle

and

\mathbf{v} = \langle 2, 3 \rangle

are parallel (one is a scalar multiple of the other).

Example 6

medium

Forces

\mathbf{F}_1 = \langle 6, 0 \rangle

N and

\mathbf{F}_2 = \langle -2, 5 \rangle

N act on a particle. Find the net force.

Example 7

hard

\mathbf{u} = \langle 3, 1 \rangle

and

\mathbf{v} = \langle -2, 4 \rangle

, express

\langle 7, 9 \rangle

a\mathbf{u} + b\mathbf{v}

Example 8

hard

Given

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

and

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

, find a unit vector in the same direction as

\mathbf{u} + \mathbf{v}

Example 9

hard

Three forces

\langle 2, 3 \rangle

\langle -4, 1 \rangle

, and

\mathbf{F}

act on a body in equilibrium. Find

\mathbf{F}

Example 10

challenge

Points

A(1, 2)

B(5, 4)

C(7, 8)

are three vertices of a parallelogram

ABCD

(in order). Find

D

using vectors.

Practice Problems

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

Example 1

easy

Compute

-3\langle 2, -4 \rangle

Example 2

medium

Find the vector from point

A(1, 3)

to point

B(4, -1)

Example 3

easy

Compute

\langle 4, 1 \rangle - \langle 1, 5 \rangle

Example 4

easy

Compute

3\langle 2, -1 \rangle

Example 5

easy

Compute

-2\langle 3, 4 \rangle

Example 6

easy

Compute

\langle 1, 2 \rangle + \langle 3, 4 \rangle

as a component sum.

Example 7

easy

Does

\langle 1, 2 \rangle - \langle 3, 4 \rangle

give a vector or a number?

Example 8

easy

Compute

\tfrac{1}{2}\langle 6, 8 \rangle

Example 9

easy

What does

\vec{u} - \vec{v}

equal in terms of addition?

Example 10

easy

Compute

2\langle 1, 1 \rangle + \langle 0, 3 \rangle

Example 11

medium

Compute

2\langle 3, -1 \rangle - 3\langle 1, 2 \rangle

Example 12

medium

Find

\vec{v}

2\vec{v} = \langle 8, -6 \rangle

Example 13

medium

Express

\langle 7, 4 \rangle

a\langle 1, 0 \rangle + b\langle 0, 1 \rangle

. Find

a, b

Example 14

medium

Compute

\langle 2, 3, 1 \rangle - 2\langle 1, 0, 1 \rangle

Example 15

medium

\langle 4, 6 \rangle

a scalar multiple of

\langle 2, 3 \rangle

? If so, what scalar?

Example 16

medium

Compute

\vec{u} + \vec{v}

where

\vec{u} = 3\langle 1, 2 \rangle

and

\vec{v} = -\langle 2, 1 \rangle

Example 17

medium

Why is

\langle 1, 2 \rangle + 3

undefined?

Example 18

medium

Compute

3\langle 2, 1 \rangle - 2\langle 1, 4 \rangle

Example 19

medium

Find

\vec{v}

3\vec{v} = \langle 9, -6 \rangle

Example 20

challenge

Find scalars

a, b

with

a\langle 1, 1 \rangle + b\langle 1, -1 \rangle = \langle 4, 2 \rangle

Example 21

challenge

Are

\langle 1, 2 \rangle

and

\langle 2, 4 \rangle

linearly independent? Explain.

Example 22

challenge

Find

\vec{v}

such that

\langle 1, 3 \rangle + 2\vec{v} = \langle 5, 1 \rangle

Example 23

easy

Compute

\langle 5, -2 \rangle + \langle -3, 7 \rangle

Example 24

easy

Compute

4\langle -1, 3 \rangle

Example 25

medium

\mathbf{a} = \langle 2, -3 \rangle

and

\mathbf{b} = \langle -1, 4 \rangle

, compute

3\mathbf{a} + 2\mathbf{b}

Example 26

medium

Given

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

and

\mathbf{v} = \langle 4, -1, 0 \rangle

, find

\mathbf{u} + 2\mathbf{v}

Example 27

medium

Find the vector

\mathbf{x}

that satisfies

\mathbf{x} + \langle 2, -5 \rangle = \langle -1, 6 \rangle

Example 28

medium

Compute

\langle 3, -2, 5 \rangle - 2\langle 1, -1, 2 \rangle

Example 29

medium

A boat travels with velocity

\langle 4, 3 \rangle

km/h relative to water, and the water moves at

\langle 1, -1 \rangle

km/h relative to ground. Find the boat's velocity relative to ground.

Example 30

medium

Find the midpoint of segment from

A(2, 6)

B(8, -2)

using the vector formula

\tfrac{1}{2}(\vec{OA} + \vec{OB})

Example 31

hard

Find scalars

a

and

b

so that

a\langle 1, 2 \rangle + b\langle 3, -1 \rangle = \langle 11, 0 \rangle

Example 32

hard

Vectors

\mathbf{u} = \langle 2, k \rangle

and

\mathbf{v} = \langle 6, -9 \rangle

are parallel. Find

k

Example 33

hard

A particle has displacement

\langle 3, -4 \rangle

m in stage 1 and

\langle -1, 2 \rangle

m in stage 2. What single displacement vector would replace both stages?

Example 34

hard

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

, find a vector parallel to

\mathbf{u}

with magnitude

7

Example 35

hard

Find a non-zero vector

\mathbf{w}

such that

\mathbf{w} = -\tfrac{1}{2}\mathbf{w} + \langle 6, -9 \rangle

Example 36

challenge

Vectors

\mathbf{a}

and

\mathbf{b}

satisfy

2\mathbf{a} + 3\mathbf{b} = \langle 1, 0 \rangle

and

\mathbf{a} - \mathbf{b} = \langle 0, 1 \rangle

. Find

\mathbf{a}

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Related Concepts

Coordinate Plane Expressions Vector Magnitude and Direction Dot Product Cross Product

Background Knowledge

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

coordinate planeexpressions