Practice Vector Addition in Math

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

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Quick Recap

Vector addition combines vectors component-wise or head-to-tail to produce a resultant vector.

Walk one arrow, then another; the single shortcut arrow is their sum.

Showing a random 20 of 50 problems.

Example 1

medium
The resultant of a⃗+b⃗\vec{a} + \vec{b} is the diagonal of what shape formed by a⃗\vec{a} and b⃗\vec{b}?

Example 2

medium
If a⃗=⟨1,2⟩\vec{a} = \langle 1, 2 \rangle and a⃗+b⃗=⟨1,2⟩\vec{a} + \vec{b} = \langle 1, 2 \rangle, what is b⃗\vec{b}?

Example 3

easy
Add ⟨−4,7⟩+⟨9,−2⟩\langle -4, 7 \rangle + \langle 9, -2 \rangle.

Example 4

challenge
Three vectors sum to zero: ⟨2,3⟩+⟨−5,1⟩+c⃗=0⃗\langle 2, 3 \rangle + \langle -5, 1 \rangle + \vec{c} = \vec{0}. Find c⃗\vec{c}.

Example 5

challenge
Let a⃗=⟨1,2⟩\vec{a}=\langle 1,2\rangle and b⃗=⟨3,−1⟩\vec{b}=\langle 3,-1\rangle. Find scalars s,ts, t such that sa⃗+tb⃗=⟨11,1⟩s\vec{a}+t\vec{b}=\langle 11, 1\rangle.

Example 6

easy
Add ⟨0,5⟩+⟨4,0⟩\langle 0, 5 \rangle + \langle 4, 0 \rangle.

Example 7

easy
Add ⟨2,3⟩+⟨2,3⟩\langle 2, 3 \rangle + \langle 2, 3 \rangle.

Example 8

easy
Add ⟨5,−3⟩+⟨−5,3⟩\langle 5, -3 \rangle + \langle -5, 3 \rangle.

Example 9

easy
Fill in the blank: a⃗+0⃗=\vec{a} + \vec{0} = ____.

Example 10

medium
Find the resultant of ⟨6,−2⟩\langle 6, -2 \rangle, ⟨−3,5⟩\langle -3, 5 \rangle, ⟨4,1⟩\langle 4, 1 \rangle, and ⟨−2,−4⟩\langle -2, -4 \rangle.

Example 11

medium
A person walks ⟨3,0⟩\langle 3, 0 \rangle then ⟨0,4⟩\langle 0, 4 \rangle. What single displacement vector results?

Example 12

hard
Find a vector c⃗\vec{c} such that ⟨2,−3,5⟩+⟨−1,4,−2⟩+c⃗=⟨0,0,0⟩\langle 2, -3, 5 \rangle + \langle -1, 4, -2 \rangle + \vec{c} = \langle 0, 0, 0 \rangle.

Example 13

challenge
If u⃗+v⃗=⟨5,7⟩\vec{u} + \vec{v} = \langle 5, 7 \rangle and u⃗−v⃗=⟨1,3⟩\vec{u} - \vec{v} = \langle 1, 3 \rangle, find u⃗\vec{u}.

Example 14

easy
What is the resultant of ⟨3,0⟩+⟨0,4⟩\langle 3, 0 \rangle + \langle 0, 4 \rangle?

Example 15

easy
Is ⟨1,2⟩+⟨3,4,5⟩\langle 1, 2 \rangle + \langle 3, 4, 5 \rangle defined?

Example 16

medium
If a⃗+b⃗=⟨10,6⟩\vec{a} + \vec{b} = \langle 10, 6 \rangle and a⃗=⟨4,9⟩\vec{a} = \langle 4, 9 \rangle, find b⃗\vec{b}.

Example 17

easy
Add ⟨−2,5⟩+⟨6,−1⟩\langle -2, 5 \rangle + \langle 6, -1 \rangle.

Example 18

easy
True or false: a⃗+b⃗=b⃗+a⃗\vec{a} + \vec{b} = \vec{b} + \vec{a} for all vectors of the same dimension.

Example 19

medium
Add ⟨4,−2⟩+⟨−1,6⟩+⟨2,1⟩\langle 4, -2 \rangle + \langle -1, 6 \rangle + \langle 2, 1 \rangle.

Example 20

medium
Find ⟨2,−1⟩+⟨5,3⟩+⟨−1,2⟩\langle 2, -1 \rangle + \langle 5, 3 \rangle + \langle -1, 2 \rangle.
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