Vector Addition Examples in Math

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

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

Concept 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.

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: Vector addition joins two vectors tip-to-tail and gives the single shortcut arrow from the start to the final end, computed by adding matching components.

Common stuck point: The procedure for vector addition is the easy part; the trap is adding the lengths instead of the components. Asking "Do I have two vectors acting together and want the single combined (resultant) vector?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do I have two vectors acting together and want the single combined (resultant) vector?

Worked Examples

Example 1

easy
Add ⟨2,1⟩+⟨−1,3⟩\langle 2, 1 \rangle + \langle -1, 3 \rangle.

Answer

⟨1,4⟩\langle 1, 4 \rangle

First step

1
Step 1: Add corresponding components: (2+(−1),1+3)(2 + (-1), 1 + 3).

Full solution

  1. 2
    Step 2: =⟨1,4⟩= \langle 1, 4 \rangle.
  2. 3
    Check: Geometrically, this is the diagonal of a parallelogram formed by the two vectors ✓
Vector addition is done component-wise: add the xx-components together and the yy-components together. Geometrically, it's the tip-to-tail method or the parallelogram diagonal.

Example 2

medium
Find u+v+w\mathbf{u} + \mathbf{v} + \mathbf{w} where u=⟨1,−2,3⟩\mathbf{u} = \langle 1, -2, 3 \rangle, v=⟨0,5,−1⟩\mathbf{v} = \langle 0, 5, -1 \rangle, w=⟨−3,1,2⟩\mathbf{w} = \langle -3, 1, 2 \rangle.

Example 3

easy
Add ⟨10,−6⟩+⟨−4,6⟩\langle 10, -6 \rangle + \langle -4, 6 \rangle component-by-component.

Example 4

medium
A drone flies ⟨5,2⟩\langle 5, 2 \rangle km, then ⟨−2,3⟩\langle -2, 3 \rangle km, then ⟨1,−4⟩\langle 1, -4 \rangle km. What is its total displacement vector?

Example 5

medium
Show that (a⃗+b⃗)+c⃗=a⃗+(b⃗+c⃗)(\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c}) for a⃗=⟨1,2⟩\vec{a}=\langle 1,2\rangle, b⃗=⟨3,4⟩\vec{b}=\langle 3,4\rangle, c⃗=⟨5,6⟩\vec{c}=\langle 5,6\rangle.

Example 6

hard
Vectors a⃗=⟨3,4⟩\vec{a}=\langle 3, 4\rangle and b⃗=⟨−4,3⟩\vec{b}=\langle -4, 3\rangle. Find a⃗+b⃗\vec{a}+\vec{b}, its magnitude, and verify it equals ∣a⃗∣2|\vec{a}|\sqrt{2}.

Practice Problems

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

Example 1

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

Example 2

medium
A boat travels ⟨4,3⟩\langle 4, 3 \rangle km then ⟨−1,5⟩\langle -1, 5 \rangle km. What is the total displacement?

Example 3

easy
Add ⟨1,2⟩+⟨3,4⟩\langle 1, 2 \rangle + \langle 3, 4 \rangle.

Example 4

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

Example 5

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

Example 6

easy
Add the zero vector: ⟨5,−3⟩+⟨0,0⟩\langle 5, -3 \rangle + \langle 0, 0 \rangle.

Example 7

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

Example 8

easy
Add ⟨7,1⟩+⟨−7,−1⟩\langle 7, 1 \rangle + \langle -7, -1 \rangle.

Example 9

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

Example 10

easy
Geometrically, how are two vectors added?

Example 11

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

Example 12

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

Example 13

medium
Find vector v⃗\vec{v} such that ⟨2,5⟩+v⃗=⟨6,1⟩\langle 2, 5 \rangle + \vec{v} = \langle 6, 1 \rangle.

Example 14

medium
Two forces ⟨4,3⟩\langle 4, 3 \rangle and ⟨−1,1⟩\langle -1, 1 \rangle act on an object. Find the net force.

Example 15

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 16

medium
Add the 3D vectors ⟨1,0,2⟩+⟨3,5,−2⟩\langle 1, 0, 2 \rangle + \langle 3, 5, -2 \rangle.

Example 17

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 18

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

Example 19

medium
Find w⃗\vec{w} if ⟨3,1⟩+w⃗=⟨0,0⟩\langle 3, 1 \rangle + \vec{w} = \langle 0, 0 \rangle.

Example 20

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 21

challenge
A boat heads ⟨0,5⟩\langle 0, 5 \rangle while a current pushes ⟨3,0⟩\langle 3, 0 \rangle. Find the resultant and its magnitude.

Example 22

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 23

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

Example 24

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

Example 25

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

Example 26

easy
Add ⟨8,3⟩+⟨8,3⟩\langle 8, 3 \rangle + \langle 8, 3 \rangle. (This is also 2⟨8,3⟩2\langle 8,3\rangle.)

Example 27

medium
Find u⃗+v⃗\vec{u} + \vec{v} where u⃗=⟨−7,4,2⟩\vec{u} = \langle -7, 4, 2 \rangle and v⃗=⟨5,−4,6⟩\vec{v} = \langle 5, -4, 6 \rangle.

Example 28

medium
Given a⃗=⟨3,−2⟩\vec{a} = \langle 3, -2 \rangle, find a⃗+a⃗+a⃗\vec{a} + \vec{a} + \vec{a}.

Example 29

medium
Solve for x⃗\vec{x}: ⟨7,−2⟩+x⃗=⟨3,5⟩\langle 7, -2 \rangle + \vec{x} = \langle 3, 5 \rangle.

Example 30

medium
Two forces F⃗1=⟨6,0⟩\vec{F}_1 = \langle 6, 0 \rangle N and F⃗2=⟨0,8⟩\vec{F}_2 = \langle 0, 8 \rangle N act on a point. Find the resultant force and its magnitude.

Example 31

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 32

medium
A particle's displacements are ⟨2,1,−1⟩\langle 2, 1, -1 \rangle, ⟨0,3,4⟩\langle 0, 3, 4 \rangle, and ⟨−1,−2,5⟩\langle -1, -2, 5 \rangle. Find the total displacement.

Example 33

medium
A swimmer's velocity relative to water is ⟨0,4⟩\langle 0, 4 \rangle m/s, and the river current is ⟨3,0⟩\langle 3, 0 \rangle m/s. Find the swimmer's velocity relative to the ground and its speed.

Example 34

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 35

hard
If u⃗+v⃗=⟨8,12⟩\vec{u} + \vec{v} = \langle 8, 12 \rangle and u⃗−v⃗=⟨2,−4⟩\vec{u} - \vec{v} = \langle 2, -4 \rangle, find u⃗\vec{u} and v⃗\vec{v}.

Example 36

hard
Three vectors of equal magnitude 66 are evenly spaced (120° apart) in the plane and start at the origin. What is their sum?

Example 37

hard
Vectors a⃗\vec{a} and b⃗\vec{b} have ∣a⃗∣=5|\vec{a}|=5 and ∣b⃗∣=12|\vec{b}|=12. The angle between them is 90°90°. Find ∣a⃗+b⃗∣|\vec{a}+\vec{b}|.

Example 38

hard
If u⃗=⟨2,k⟩\vec{u} = \langle 2, k \rangle and v⃗=⟨k,8⟩\vec{v} = \langle k, 8 \rangle, find kk such that u⃗+v⃗=⟨5,11⟩\vec{u} + \vec{v} = \langle 5, 11 \rangle.

Example 39

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 40

challenge
nn unit vectors point outward from the origin to the vertices of a regular nn-gon. What is their sum?

Example 41

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.
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Background Knowledge

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

vector intuitionvector operationsdisplacement geometric