Practice Vector Addition, Subtraction, and Scalar Multiplication in Math

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

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

medium

True or false:

k(\mathbf{u} + \mathbf{v}) = k\mathbf{u} + k\mathbf{v}

for any scalar

k

Example 2

challenge

Points

A(1, 2)

B(5, 4)

C(7, 8)

are three vertices of a parallelogram

ABCD

(in order). Find

D

using vectors.

Example 3

hard

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

, find a vector parallel to

\mathbf{u}

with magnitude

7

Example 4

easy

Compute

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

Example 5

easy

Compute

4\langle -1, 3 \rangle

Example 6

medium

Why is

\langle 1, 2 \rangle + 3

undefined?

Example 7

easy

Compute

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

as a component sum.

Example 8

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 9

medium

Find the vector

\mathbf{x}

that satisfies

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

Example 10

easy

Compute

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

Example 11

easy

What is the sum

\mathbf{u} + (-\mathbf{u})

for any vector

\mathbf{u}

Example 12

medium

Given

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

and

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

, find

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

Example 13

medium

Fill in:

\langle a, b \rangle + \langle c, d \rangle = \langle\, \underline{\quad}\, , \, \underline{\quad}\, \rangle

Example 14

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 15

challenge

Are

\langle 1, 2 \rangle

and

\langle 2, 4 \rangle

linearly independent? Explain.

Example 16

medium

Find

\vec{v}

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

Example 17

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 18

medium

Find scalars

s

and

t

so that

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

Example 19

challenge

Find scalars

a, b

with

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

Example 20

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