Cross Product Examples: 45 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Cross 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 cross product of two 3D vectors $\mathbf{a} = \langle a_1, a_2, a_3 \rangle$ and $\mathbf{b} = \langle b_1, b_2, b_3 \rangle$ is a new vector $\mathbf{a} \times \mathbf{b}$ that is perpendicular to both $\mathbf{a}$ and $\mathbf{b}$ . Its magnitude equals the area of the parallelogram formed by $\mathbf{a}$ and $\mathbf{b}$ .

Place two arrows flat on a table. The cross product points straight up from the table, perpendicular to both. Its length tells you how much area the two arrows span—like the area of a parallelogram with the arrows as sides. If the arrows are parallel, they span no area, so the cross product is the zero vector.

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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 cross product of two 3D vectors returns a new vector perpendicular to both, whose length is the area of the parallelogram they span.

Common stuck point: The procedure for cross product is the easy part; the trap is returning a scalar. Asking "Do I have two 3D vectors and need a new vector perpendicular to both (or the area they span)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do I have two 3D vectors and need a new vector perpendicular to both (or the area they span)?

Worked Examples

Example 1

medium

Find

\langle 1, 0, 0 \rangle \times \langle 0, 1, 0 \rangle

Answer

\langle 0, 0, 1 \rangle

First step

Step 1: Use the formula:

\mathbf{a} \times \mathbf{b} = \langle a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1 \rangle

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Example 2

hard

Find

\langle 2, 3, 1 \rangle \times \langle 1, -1, 2 \rangle

Example 3

easy

Show the pattern: compute the third component of

\langle a_1,a_2,0\rangle\times\langle b_1,b_2,0\rangle

Example 4

medium

Find a normal vector to the plane through

P=(1,0,0)

Q=(0,1,0)

R=(0,0,1)

Example 5

medium

Why must you embed 2D vectors

\langle 2,3\rangle

and

\langle 5,1\rangle

as 3D before cross-producting?

Example 6

hard

Find the volume of the parallelepiped spanned by

\mathbf{u}=\langle 1,0,2\rangle

\mathbf{v}=\langle 0,3,0\rangle

\mathbf{w}=\langle 1,1,4\rangle

Example 7

challenge

Prove the Jacobi-like identity:

\mathbf{a}\times(\mathbf{b}\times\mathbf{c}) = \mathbf{b}(\mathbf{a}\cdot\mathbf{c}) - \mathbf{c}(\mathbf{a}\cdot\mathbf{b})

on the test case

\mathbf{a}=\hat{i}

\mathbf{b}=\hat{j}

\mathbf{c}=\hat{k}

Practice Problems

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

Example 1

medium

Find

\langle 1, 2, 0 \rangle \times \langle 3, 0, 0 \rangle

Example 2

easy

\mathbf{a} \times \mathbf{b} = \mathbf{b} \times \mathbf{a}

Example 3

easy

Compute

\langle 1, 0, 0 \rangle \times \langle 0, 1, 0 \rangle

Example 4

easy

Compute

\langle 0, 1, 0 \rangle \times \langle 1, 0, 0 \rangle

Example 5

easy

Compute

\langle 2, 0, 0 \rangle \times \langle 0, 3, 0 \rangle

Example 6

easy

\langle 0,0,5\rangle

perpendicular to both

\langle1,0,0\rangle

and

\langle0,1,0\rangle

Example 7

easy

Compute

\langle 1, 2, 3 \rangle \times \langle 1, 2, 3 \rangle

Example 8

easy

Compute the third (z) component of

\langle 3, 1, 0 \rangle \times \langle 2, 5, 0 \rangle

Example 9

easy

Extend the 2D vectors

\langle 4, 1 \rangle

and

\langle 2, 3 \rangle

to 3D and give them in

\langle\cdot,\cdot,0\rangle

form.

Example 10

easy

What does the magnitude of

\mathbf{a}\times\mathbf{b}

represent geometrically?

Example 11

medium

Compute

\langle 2, 3, 4 \rangle \times \langle 5, 6, 7 \rangle

Example 12

medium

Compute

\langle 1, 0, 2 \rangle \times \langle 3, 1, 0 \rangle

Example 13

medium

Find the area of the parallelogram spanned by

\langle 3, 0, 0 \rangle

and

\langle 0, 4, 0 \rangle

Example 14

medium

Find the area of the triangle with vertices

A=(0,0,0)

B=(2,0,0)

C=(0,3,0)

Example 15

medium

Verify that

\langle 2,3,4\rangle\times\langle5,6,7\rangle = \langle -3,6,-3\rangle

is perpendicular to

\langle 2,3,4\rangle

Example 16

medium

Find

\mathbf{b}\times\mathbf{a}

given

\mathbf{a}\times\mathbf{b}=\langle -3, 6, -3 \rangle

Example 17

medium

Two vectors have magnitudes

3

and

5

with a

30^\circ

angle between them. Find

\|\mathbf{a}\times\mathbf{b}\|

Example 18

medium

For what

c

\langle 1, 2, c \rangle \times \langle 2, 4, 6 \rangle = \langle 0,0,0\rangle

Example 19

challenge

Find a unit vector perpendicular to both

\langle 1, 0, 1 \rangle

and

\langle 0, 1, 1 \rangle

Example 20

challenge

Find the volume of the parallelepiped spanned by

\langle1,0,0\rangle

\langle1,1,0\rangle

\langle1,1,1\rangle

using the scalar triple product.

Example 21

challenge

Show that

\mathbf{a}\times\mathbf{a}=\mathbf{0}

for any vector

\mathbf{a}=\langle a_1,a_2,a_3\rangle

Example 22

medium

Compute

\langle 2, -1, 3 \rangle \times \langle 0, 4, -2 \rangle

Example 23

easy

Compute

\langle 0, 0, 1 \rangle \times \langle 1, 0, 0 \rangle

Example 24

easy

Find

\langle 3, 0, 0 \rangle \times \langle 0, 0, 2 \rangle

Example 25

easy

True or false:

\mathbf{a}\times\mathbf{b}

is perpendicular to

\mathbf{a}

Example 26

medium

Compute

\langle 1, 2, 3 \rangle \times \langle 4, 5, 6 \rangle

Example 27

medium

Find

\langle 0, 3, -1 \rangle \times \langle 2, 1, 4 \rangle

Example 28

medium

Find the area of the parallelogram with adjacent sides

\mathbf{u}=\langle 1,2,2\rangle

and

\mathbf{v}=\langle 3,0,0\rangle

Example 29

medium

Find the area of the triangle with vertices

A=(1,1,0)

B=(4,1,0)

C=(1,5,0)

Example 30

medium

Find

\langle 2, -1, 0 \rangle \times \langle 0, 0, 5 \rangle

Example 31

medium

For

\mathbf{a}=\langle 1,1,1\rangle

and

\mathbf{b}=\langle 2,2,2\rangle

, what is

\mathbf{a}\times\mathbf{b}

Example 32

medium

Find

\langle -1, 2, 3 \rangle \times \langle 4, 0, -2 \rangle

Example 33

hard

Find a unit vector perpendicular to both

\langle 2, 0, 1 \rangle

and

\langle 0, 3, 1 \rangle

Example 34

hard

Find the angle between

\mathbf{a}=\langle 1,0,0\rangle

and

\mathbf{b}=\langle 1,1,0\rangle

using the magnitude of the cross product.

Example 35

hard

Compute the scalar triple product

\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})

for

\mathbf{a}=\langle 1,2,3\rangle

\mathbf{b}=\langle 0,1,0\rangle

\mathbf{c}=\langle 0,0,1\rangle

Example 36

hard

Find the equation of the plane through

(1,1,1)

with normal

\langle 1,0,1\rangle\times\langle 0,1,1\rangle

Example 37

hard

Compute

\langle 3, 1, -2 \rangle \times \langle 1, -1, 1 \rangle

Example 38

challenge

For

\mathbf{a}=\langle 1,2,2\rangle

and

\mathbf{b}=\langle 2,1,-2\rangle

, find the area of the parallelogram and verify

\|\mathbf{a}\|\|\mathbf{b}\|\sin\theta

matches.

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Related Concepts

Dot Product Vector Addition, Subtraction, and Scalar Multiplication Determinant Surface Area

Background Knowledge

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

dot productvector operationsdeterminant