Dot Product Examples in Math

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

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

Concept Recap

The dot product of two vectors a=⟨a1,a2⟩\mathbf{a} = \langle a_1, a_2 \rangle and b=⟨b1,b2⟩\mathbf{b} = \langle b_1, b_2 \rangle is the scalar a⋅b=a1b1+a2b2\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2. Equivalently, a⋅b=∥a∥∥b∥cos⁡θ\mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos\theta, where θ\theta is the angle between the vectors.

The dot product measures how much two vectors point in the same direction. If they point the same way, the dot product is large and positive. If perpendicular, it is zero. If they point in opposite directions, it is negative. Think of it as a 'similarity score' for directions.

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: The dot product multiplies matching components and adds, giving one number that is positive when vectors point alike, zero when perpendicular, negative when opposed.

Common stuck point: The procedure for dot product is the easy part; the trap is reporting a vector as the answer. Asking "Do I have two vectors and need one number measuring their directional agreement (not a new 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 and need one number measuring their directional agreement (not a new vector)?

Worked Examples

Example 1

easy
Find ⟨1,2⟩⋅⟨3,−1⟩\langle 1, 2 \rangle \cdot \langle 3, -1 \rangle.

Answer

11

First step

1
Step 1: Multiply corresponding components and sum: 1(3)+2(−1)1(3) + 2(-1).

Full solution

  1. 2
    Step 2: =3−2=1= 3 - 2 = 1.
  2. 3
    Note: The result is a scalar (number), not a vector.
The dot product multiplies corresponding components and sums the results, producing a scalar. A positive dot product means the vectors point in roughly the same direction (angle < 90°).

Example 2

medium
Find the angle between a=⟨1,0⟩\mathbf{a} = \langle 1, 0 \rangle and b=⟨1,1⟩\mathbf{b} = \langle 1, 1 \rangle.

Example 3

easy
Compute ⟨2,5⟩⋅⟨2,5⟩\langle 2, 5 \rangle \cdot \langle 2, 5 \rangle and interpret it.

Example 4

medium
Use the distributive property to expand (a+b)⋅c(\mathbf{a} + \mathbf{b}) \cdot \mathbf{c} for a=⟨1,2⟩\mathbf{a} = \langle 1, 2 \rangle, b=⟨3,0⟩\mathbf{b} = \langle 3, 0 \rangle, c=⟨4,−1⟩\mathbf{c} = \langle 4, -1 \rangle.

Example 5

hard
Show that (a−b)⋅(a+b)=∥a∥2−∥b∥2(\mathbf{a} - \mathbf{b}) \cdot (\mathbf{a} + \mathbf{b}) = \|\mathbf{a}\|^2 - \|\mathbf{b}\|^2.

Practice Problems

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

Example 1

easy
Are ⟨2,3⟩\langle 2, 3 \rangle and ⟨−3,2⟩\langle -3, 2 \rangle perpendicular?

Example 2

hard
Find ⟨2,−1,3⟩⋅⟨4,5,−2⟩\langle 2, -1, 3 \rangle \cdot \langle 4, 5, -2 \rangle.

Example 3

easy
Compute ⟨2,3⟩⋅⟨4,1⟩\langle 2, 3 \rangle \cdot \langle 4, 1 \rangle.

Example 4

easy
Compute ⟨1,0⟩⋅⟨0,1⟩\langle 1, 0 \rangle \cdot \langle 0, 1 \rangle.

Example 5

easy
Compute ⟨5,2⟩⋅⟨3,4⟩\langle 5, 2 \rangle \cdot \langle 3, 4 \rangle.

Example 6

easy
Compute ⟨−2,3⟩⋅⟨4,−1⟩\langle -2, 3 \rangle \cdot \langle 4, -1 \rangle.

Example 7

easy
Compute ⟨1,2,3⟩⋅⟨4,5,6⟩\langle 1, 2, 3 \rangle \cdot \langle 4, 5, 6 \rangle.

Example 8

easy
Are ⟨3,6⟩\langle 3, 6 \rangle and ⟨−2,1⟩\langle -2, 1 \rangle perpendicular?

Example 9

easy
Compute ⟨0,0⟩⋅⟨7,9⟩\langle 0, 0 \rangle \cdot \langle 7, 9 \rangle.

Example 10

easy
Compute ⟨4,4⟩⋅⟨4,4⟩\langle 4, 4 \rangle \cdot \langle 4, 4 \rangle.

Example 11

medium
Find kk so that ⟨2,k⟩\langle 2, k \rangle and ⟨6,−3⟩\langle 6, -3 \rangle are perpendicular.

Example 12

medium
Use a⋅b=∥a∥∥b∥cos⁡θ\mathbf{a}\cdot\mathbf{b}=\|\mathbf{a}\|\|\mathbf{b}\|\cos\theta to find cos⁡θ\cos\theta for a=⟨3,4⟩\mathbf{a}=\langle 3,4\rangle, b=⟨4,3⟩\mathbf{b}=\langle 4,3\rangle.

Example 13

medium
Compute (2a)⋅b(2\mathbf{a})\cdot\mathbf{b} where a=⟨1,2⟩\mathbf{a}=\langle 1,2\rangle, b=⟨3,4⟩\mathbf{b}=\langle 3,4\rangle.

Example 14

medium
Compute a⋅(b+c)\mathbf{a}\cdot(\mathbf{b}+\mathbf{c}) for a=⟨2,1⟩\mathbf{a}=\langle 2,1\rangle, b=⟨1,3⟩\mathbf{b}=\langle 1,3\rangle, c=⟨4,−1⟩\mathbf{c}=\langle 4,-1\rangle.

Example 15

medium
Is the angle between ⟨1,2⟩\langle 1, 2 \rangle and ⟨3,−1⟩\langle 3, -1 \rangle acute, right, or obtuse?

Example 16

medium
Find the projection length of a=⟨4,3⟩\mathbf{a}=\langle 4,3\rangle onto b=⟨1,0⟩\mathbf{b}=\langle 1,0\rangle.

Example 17

medium
Compute the work done by force F=⟨3,4⟩\mathbf{F}=\langle 3, 4 \rangle over displacement d=⟨2,1⟩\mathbf{d}=\langle 2, 1 \rangle.

Example 18

medium
For what value of tt is ⟨t,2⟩⋅⟨t,−8⟩=0\langle t, 2 \rangle \cdot \langle t, -8 \rangle = 0?

Example 19

challenge
Vectors a=⟨1,2⟩\mathbf{a}=\langle 1,2\rangle and b=⟨3,c⟩\mathbf{b}=\langle 3, c\rangle make a 45∘45^\circ angle. The dot product is positive. Show the condition a⋅b>0\mathbf{a}\cdot\mathbf{b}>0 in terms of cc.

Example 20

challenge
Prove that for any vectors, ∥a+b∥2=∥a∥2+2(a⋅b)+∥b∥2\|\mathbf{a}+\mathbf{b}\|^2 = \|\mathbf{a}\|^2 + 2(\mathbf{a}\cdot\mathbf{b}) + \|\mathbf{b}\|^2.

Example 21

challenge
Three points A=(0,0)A=(0,0), B=(4,0)B=(4,0), C=(1,3)C=(1,3) form a triangle. Use the dot product to find the angle at AA (give cos⁡\cos).

Example 22

medium
Compute ⟨7,−2,1⟩⋅⟨1,3,−4⟩\langle 7, -2, 1 \rangle \cdot \langle 1, 3, -4 \rangle.

Example 23

easy
Compute ⟨3,4⟩⋅⟨1,2⟩\langle 3, 4 \rangle \cdot \langle 1, 2 \rangle.

Example 24

easy
Compute ⟨−1,5⟩⋅⟨2,3⟩\langle -1, 5 \rangle \cdot \langle 2, 3 \rangle.

Example 25

easy
Are ⟨4,1⟩\langle 4, 1 \rangle and ⟨−1,4⟩\langle -1, 4 \rangle perpendicular?

Example 26

easy
Compute ⟨1,1,1⟩⋅⟨2,3,4⟩\langle 1, 1, 1 \rangle \cdot \langle 2, 3, 4 \rangle.

Example 27

easy
Compute ⟨−3,6⟩⋅⟨4,−2⟩\langle -3, 6 \rangle \cdot \langle 4, -2 \rangle.

Example 28

medium
Find kk so that ⟨5,k⟩\langle 5, k \rangle and ⟨2,3⟩\langle 2, 3 \rangle are perpendicular.

Example 29

medium
Find cos⁡θ\cos\theta for a=⟨1,1⟩\mathbf{a} = \langle 1, 1\rangle and b=⟨1,0⟩\mathbf{b} = \langle 1, 0\rangle.

Example 30

medium
Given a=⟨3,4⟩\mathbf{a} = \langle 3, 4 \rangle and b=⟨4,−3⟩\mathbf{b} = \langle 4, -3 \rangle, find the angle between them.

Example 31

medium
Find the scalar projection of a=⟨5,2⟩\mathbf{a} = \langle 5, 2 \rangle onto b=⟨1,0⟩\mathbf{b} = \langle 1, 0 \rangle.

Example 32

medium
Is the angle between ⟨2,5⟩\langle 2, 5 \rangle and ⟨6,−1⟩\langle 6, -1 \rangle acute, right, or obtuse?

Example 33

medium
Compute the work done by force F=⟨5,12⟩\mathbf{F} = \langle 5, 12 \rangle over displacement d=⟨3,4⟩\mathbf{d} = \langle 3, 4 \rangle.

Example 34

medium
Find kk so that ⟨k,4⟩\langle k, 4 \rangle is perpendicular to ⟨8,k⟩\langle 8, k \rangle.

Example 35

medium
Find the vector projection of a=⟨3,4⟩\mathbf{a} = \langle 3, 4 \rangle onto b=⟨1,0⟩\mathbf{b} = \langle 1, 0 \rangle.

Example 36

hard
Given ∥a∥=3\|\mathbf{a}\| = 3, ∥b∥=5\|\mathbf{b}\| = 5, and the angle between them is 60∘60^\circ, find a⋅b\mathbf{a} \cdot \mathbf{b}.

Example 37

hard
If ∥a∥=4\|\mathbf{a}\| = 4 and ∥b∥=3\|\mathbf{b}\| = 3 and a⋅b=6\mathbf{a} \cdot \mathbf{b} = 6, find ∥a+b∥\|\mathbf{a} + \mathbf{b}\|.

Example 38

hard
Find a unit vector perpendicular to ⟨3,4⟩\langle 3, 4 \rangle.

Example 39

hard
Triangle has vertices A=(1,1)A = (1, 1), B=(5,1)B = (5, 1), C=(1,4)C = (1, 4). Use the dot product to confirm the angle at AA.

Example 40

hard
Compute ⟨2,−1,4⟩⋅⟨3,6,2⟩\langle 2, -1, 4 \rangle \cdot \langle 3, 6, 2 \rangle.

Example 41

hard
For what value(s) of tt is the angle between ⟨1,t⟩\langle 1, t \rangle and ⟨t,4⟩\langle t, 4 \rangle acute?

Example 42

challenge
Vectors a\mathbf{a} and b\mathbf{b} satisfy ∥a∥=∥b∥=1\|\mathbf{a}\| = \|\mathbf{b}\| = 1 and ∥a−b∥=1\|\mathbf{a} - \mathbf{b}\| = 1. Find a⋅b\mathbf{a} \cdot \mathbf{b}.
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Background Knowledge

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

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