Practice Dot Product in Math

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

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

easy
Compute ⟨0,0,0⟩⋅⟨5,−7,9⟩\langle 0, 0, 0 \rangle \cdot \langle 5, -7, 9 \rangle.

Example 2

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

Example 3

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

Example 4

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

Example 5

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

Example 6

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

Example 7

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

Example 8

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

Example 9

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 10

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.

Example 11

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

Example 12

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

Example 13

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

Example 14

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

Example 15

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

Example 16

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

Example 17

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}.

Example 18

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 19

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

Example 20

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.
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