Vector Addition, Subtraction, and Scalar Multiplication Examples in Math

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 u=⟨u1,u2⟩\mathbf{u} = \langle u_1, u_2 \rangle and v=⟨v1,v2⟩\mathbf{v} = \langle v_1, v_2 \rangle, then u+v=⟨u1+v1,u2+v2⟩\mathbf{u} + \mathbf{v} = \langle u_1 + v_1, u_2 + v_2 \rangle and ku=⟨ku1,ku2⟩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: 2v2\mathbf{v} is twice as long in the same direction, while −v-\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
If u=⟨3,−1⟩\mathbf{u} = \langle 3, -1 \rangle and v=⟨1,4⟩\mathbf{v} = \langle 1, 4 \rangle, find 2u−v2\mathbf{u} - \mathbf{v}.

Answer

⟨5,−6⟩\langle 5, -6 \rangle

First step

1
Step 1: 2u=⟨6,−2⟩2\mathbf{u} = \langle 6, -2 \rangle.

Full solution

  1. 2
    Step 2: 2u−v=⟨6−1,−2−4⟩=⟨5,−6⟩2\mathbf{u} - \mathbf{v} = \langle 6 - 1, -2 - 4 \rangle = \langle 5, -6 \rangle.
  2. 3
    Check: Each component is 2ui−vi2u_i - v_i ✓
Scalar multiplication scales each component, then subtraction is done component-wise. Linear combinations of vectors like au+bva\mathbf{u} + b\mathbf{v} are fundamental in linear algebra.

Example 2

medium
Find u−v\mathbf{u} - \mathbf{v} where u=⟨2,5,−1⟩\mathbf{u} = \langle 2, 5, -1 \rangle and v=⟨4,−3,2⟩\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 ss and tt so that s⟨1,0⟩+t⟨0,1⟩=⟨7,−4⟩s\langle 1, 0 \rangle + t\langle 0, 1 \rangle = \langle 7, -4 \rangle.

Example 5

medium
Show that u=⟨4,6⟩\mathbf{u} = \langle 4, 6 \rangle and v=⟨2,3⟩\mathbf{v} = \langle 2, 3 \rangle are parallel (one is a scalar multiple of the other).

Example 6

medium
Forces F1=⟨6,0⟩\mathbf{F}_1 = \langle 6, 0 \rangle N and F2=⟨−2,5⟩\mathbf{F}_2 = \langle -2, 5 \rangle N act on a particle. Find the net force.

Example 7

hard
If u=⟨3,1⟩\mathbf{u} = \langle 3, 1 \rangle and v=⟨−2,4⟩\mathbf{v} = \langle -2, 4 \rangle, express ⟨7,9⟩\langle 7, 9 \rangle as au+bva\mathbf{u} + b\mathbf{v}.

Example 8

hard
Given u=⟨1,−2⟩\mathbf{u} = \langle 1, -2 \rangle and v=⟨3,1⟩\mathbf{v} = \langle 3, 1 \rangle, find a unit vector in the same direction as u+v\mathbf{u} + \mathbf{v}.

Example 9

hard
Three forces ⟨2,3⟩\langle 2, 3 \rangle, ⟨−4,1⟩\langle -4, 1 \rangle, and F\mathbf{F} act on a body in equilibrium. Find F\mathbf{F}.

Example 10

challenge
Points A(1,2)A(1, 2), B(5,4)B(5, 4), C(7,8)C(7, 8) are three vertices of a parallelogram ABCDABCD (in order). Find DD using vectors.

Practice Problems

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

Example 1

easy
Compute −3⟨2,−4⟩-3\langle 2, -4 \rangle.

Example 2

medium
Find the vector from point A(1,3)A(1, 3) to point B(4,−1)B(4, -1).

Example 3

easy
Compute ⟨4,1⟩−⟨1,5⟩\langle 4, 1 \rangle - \langle 1, 5 \rangle.

Example 4

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

Example 5

easy
Compute −2⟨3,4⟩-2\langle 3, 4 \rangle.

Example 6

easy
Compute ⟨1,2⟩+⟨3,4⟩\langle 1, 2 \rangle + \langle 3, 4 \rangle as a component sum.

Example 7

easy
Does ⟨1,2⟩−⟨3,4⟩\langle 1, 2 \rangle - \langle 3, 4 \rangle give a vector or a number?

Example 8

easy
Compute 12⟨6,8⟩\tfrac{1}{2}\langle 6, 8 \rangle.

Example 9

easy
What does u⃗−v⃗\vec{u} - \vec{v} equal in terms of addition?

Example 10

easy
Compute 2⟨1,1⟩+⟨0,3⟩2\langle 1, 1 \rangle + \langle 0, 3 \rangle.

Example 11

medium
Compute 2⟨3,−1⟩−3⟨1,2⟩2\langle 3, -1 \rangle - 3\langle 1, 2 \rangle.

Example 12

medium
Find v⃗\vec{v} if 2v⃗=⟨8,−6⟩2\vec{v} = \langle 8, -6 \rangle.

Example 13

medium
Express ⟨7,4⟩\langle 7, 4 \rangle as a⟨1,0⟩+b⟨0,1⟩a\langle 1, 0 \rangle + b\langle 0, 1 \rangle. Find a,ba, b.

Example 14

medium
Compute ⟨2,3,1⟩−2⟨1,0,1⟩\langle 2, 3, 1 \rangle - 2\langle 1, 0, 1 \rangle.

Example 15

medium
Is ⟨4,6⟩\langle 4, 6 \rangle a scalar multiple of ⟨2,3⟩\langle 2, 3 \rangle? If so, what scalar?

Example 16

medium
Compute u⃗+v⃗\vec{u} + \vec{v} where u⃗=3⟨1,2⟩\vec{u} = 3\langle 1, 2 \rangle and v⃗=−⟨2,1⟩\vec{v} = -\langle 2, 1 \rangle.

Example 17

medium
Why is ⟨1,2⟩+3\langle 1, 2 \rangle + 3 undefined?

Example 18

medium
Compute 3⟨2,1⟩−2⟨1,4⟩3\langle 2, 1 \rangle - 2\langle 1, 4 \rangle.

Example 19

medium
Find v⃗\vec{v} if 3v⃗=⟨9,−6⟩3\vec{v} = \langle 9, -6 \rangle.

Example 20

challenge
Find scalars a,ba, b with a⟨1,1⟩+b⟨1,−1⟩=⟨4,2⟩a\langle 1, 1 \rangle + b\langle 1, -1 \rangle = \langle 4, 2 \rangle.

Example 21

challenge
Are ⟨1,2⟩\langle 1, 2 \rangle and ⟨2,4⟩\langle 2, 4 \rangle linearly independent? Explain.

Example 22

challenge
Find v⃗\vec{v} such that ⟨1,3⟩+2v⃗=⟨5,1⟩\langle 1, 3 \rangle + 2\vec{v} = \langle 5, 1 \rangle.

Example 23

easy
Compute ⟨5,−2⟩+⟨−3,7⟩\langle 5, -2 \rangle + \langle -3, 7 \rangle.

Example 24

easy
Compute 4⟨−1,3⟩4\langle -1, 3 \rangle.

Example 25

medium
If a=⟨2,−3⟩\mathbf{a} = \langle 2, -3 \rangle and b=⟨−1,4⟩\mathbf{b} = \langle -1, 4 \rangle, compute 3a+2b3\mathbf{a} + 2\mathbf{b}.

Example 26

medium
Given u=⟨1,2,3⟩\mathbf{u} = \langle 1, 2, 3 \rangle and v=⟨4,−1,0⟩\mathbf{v} = \langle 4, -1, 0 \rangle, find u+2v\mathbf{u} + 2\mathbf{v}.

Example 27

medium
Find the vector x\mathbf{x} that satisfies x+⟨2,−5⟩=⟨−1,6⟩\mathbf{x} + \langle 2, -5 \rangle = \langle -1, 6 \rangle.

Example 28

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

Example 29

medium
A boat travels with velocity ⟨4,3⟩\langle 4, 3 \rangle km/h relative to water, and the water moves at ⟨1,−1⟩\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)A(2, 6) to B(8,−2)B(8, -2) using the vector formula 12(OA⃗+OB⃗)\tfrac{1}{2}(\vec{OA} + \vec{OB}).

Example 31

hard
Find scalars aa and bb so that a⟨1,2⟩+b⟨3,−1⟩=⟨11,0⟩a\langle 1, 2 \rangle + b\langle 3, -1 \rangle = \langle 11, 0 \rangle.

Example 32

hard
Vectors u=⟨2,k⟩\mathbf{u} = \langle 2, k \rangle and v=⟨6,−9⟩\mathbf{v} = \langle 6, -9 \rangle are parallel. Find kk.

Example 33

hard
A particle has displacement ⟨3,−4⟩\langle 3, -4 \rangle m in stage 1 and ⟨−1,2⟩\langle -1, 2 \rangle m in stage 2. What single displacement vector would replace both stages?

Example 34

hard
If u=⟨2,−1,3⟩\mathbf{u} = \langle 2, -1, 3 \rangle, find a vector parallel to u\mathbf{u} with magnitude 77.

Example 35

hard
Find a non-zero vector w\mathbf{w} such that w=−12w+⟨6,−9⟩\mathbf{w} = -\tfrac{1}{2}\mathbf{w} + \langle 6, -9 \rangle.

Example 36

challenge
Vectors a\mathbf{a} and b\mathbf{b} satisfy 2a+3b=⟨1,0⟩2\mathbf{a} + 3\mathbf{b} = \langle 1, 0 \rangle and a−b=⟨0,1⟩\mathbf{a} - \mathbf{b} = \langle 0, 1 \rangle. Find a\mathbf{a}.
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Background Knowledge

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

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