Cross Product Examples in Math

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 a=⟨a1,a2,a3⟩\mathbf{a} = \langle a_1, a_2, a_3 \rangle and b=⟨b1,b2,b3⟩\mathbf{b} = \langle b_1, b_2, b_3 \rangle is a new vector aΓ—b\mathbf{a} \times \mathbf{b} that is perpendicular to both a\mathbf{a} and b\mathbf{b}. Its magnitude equals the area of the parallelogram formed by a\mathbf{a} and b\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 spanβ€”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 ⟨1,0,0βŸ©Γ—βŸ¨0,1,0⟩\langle 1, 0, 0 \rangle \times \langle 0, 1, 0 \rangle.

Answer

⟨0,0,1⟩\langle 0, 0, 1 \rangle

First step

1
Step 1: Use the formula: aΓ—b=⟨a2b3βˆ’a3b2,a3b1βˆ’a1b3,a1b2βˆ’a2b1⟩\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 ⟨2,3,1βŸ©Γ—βŸ¨1,βˆ’1,2⟩\langle 2, 3, 1 \rangle \times \langle 1, -1, 2 \rangle.

Example 3

easy
Show the pattern: compute the third component of ⟨a1,a2,0βŸ©Γ—βŸ¨b1,b2,0⟩\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)P=(1,0,0), Q=(0,1,0)Q=(0,1,0), R=(0,0,1)R=(0,0,1).

Example 5

medium
Why must you embed 2D vectors ⟨2,3⟩\langle 2,3\rangle and ⟨5,1⟩\langle 5,1\rangle as 3D before cross-producting?

Example 6

hard
Find the volume of the parallelepiped spanned by u=⟨1,0,2⟩\mathbf{u}=\langle 1,0,2\rangle, v=⟨0,3,0⟩\mathbf{v}=\langle 0,3,0\rangle, w=⟨1,1,4⟩\mathbf{w}=\langle 1,1,4\rangle.

Example 7

challenge
Prove the Jacobi-like identity: aΓ—(bΓ—c)=b(aβ‹…c)βˆ’c(aβ‹…b)\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 a=i^\mathbf{a}=\hat{i}, b=j^\mathbf{b}=\hat{j}, c=k^\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 ⟨1,2,0βŸ©Γ—βŸ¨3,0,0⟩\langle 1, 2, 0 \rangle \times \langle 3, 0, 0 \rangle.

Example 2

easy
Is aΓ—b=bΓ—a\mathbf{a} \times \mathbf{b} = \mathbf{b} \times \mathbf{a}?

Example 3

easy
Compute ⟨1,0,0βŸ©Γ—βŸ¨0,1,0⟩\langle 1, 0, 0 \rangle \times \langle 0, 1, 0 \rangle.

Example 4

easy
Compute ⟨0,1,0βŸ©Γ—βŸ¨1,0,0⟩\langle 0, 1, 0 \rangle \times \langle 1, 0, 0 \rangle.

Example 5

easy
Compute ⟨2,0,0βŸ©Γ—βŸ¨0,3,0⟩\langle 2, 0, 0 \rangle \times \langle 0, 3, 0 \rangle.

Example 6

easy
Is ⟨0,0,5⟩\langle 0,0,5\rangle perpendicular to both ⟨1,0,0⟩\langle1,0,0\rangle and ⟨0,1,0⟩\langle0,1,0\rangle?

Example 7

easy
Compute ⟨1,2,3βŸ©Γ—βŸ¨1,2,3⟩\langle 1, 2, 3 \rangle \times \langle 1, 2, 3 \rangle.

Example 8

easy
Compute the third (z) component of ⟨3,1,0βŸ©Γ—βŸ¨2,5,0⟩\langle 3, 1, 0 \rangle \times \langle 2, 5, 0 \rangle.

Example 9

easy
Extend the 2D vectors ⟨4,1⟩\langle 4, 1 \rangle and ⟨2,3⟩\langle 2, 3 \rangle to 3D and give them in βŸ¨β‹…,β‹…,0⟩\langle\cdot,\cdot,0\rangle form.

Example 10

easy
What does the magnitude of aΓ—b\mathbf{a}\times\mathbf{b} represent geometrically?

Example 11

medium
Compute ⟨2,3,4βŸ©Γ—βŸ¨5,6,7⟩\langle 2, 3, 4 \rangle \times \langle 5, 6, 7 \rangle.

Example 12

medium
Compute ⟨1,0,2βŸ©Γ—βŸ¨3,1,0⟩\langle 1, 0, 2 \rangle \times \langle 3, 1, 0 \rangle.

Example 13

medium
Find the area of the parallelogram spanned by ⟨3,0,0⟩\langle 3, 0, 0 \rangle and ⟨0,4,0⟩\langle 0, 4, 0 \rangle.

Example 14

medium
Find the area of the triangle with vertices A=(0,0,0)A=(0,0,0), B=(2,0,0)B=(2,0,0), C=(0,3,0)C=(0,3,0).

Example 15

medium
Verify that ⟨2,3,4βŸ©Γ—βŸ¨5,6,7⟩=βŸ¨βˆ’3,6,βˆ’3⟩\langle 2,3,4\rangle\times\langle5,6,7\rangle = \langle -3,6,-3\rangle is perpendicular to ⟨2,3,4⟩\langle 2,3,4\rangle.

Example 16

medium
Find bΓ—a\mathbf{b}\times\mathbf{a} given aΓ—b=βŸ¨βˆ’3,6,βˆ’3⟩\mathbf{a}\times\mathbf{b}=\langle -3, 6, -3 \rangle.

Example 17

medium
Two vectors have magnitudes 33 and 55 with a 30∘30^\circ angle between them. Find βˆ₯aΓ—bβˆ₯\|\mathbf{a}\times\mathbf{b}\|.

Example 18

medium
For what cc is ⟨1,2,cβŸ©Γ—βŸ¨2,4,6⟩=⟨0,0,0⟩\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 ⟨1,0,1⟩\langle 1, 0, 1 \rangle and ⟨0,1,1⟩\langle 0, 1, 1 \rangle.

Example 20

challenge
Find the volume of the parallelepiped spanned by ⟨1,0,0⟩\langle1,0,0\rangle, ⟨1,1,0⟩\langle1,1,0\rangle, ⟨1,1,1⟩\langle1,1,1\rangle using the scalar triple product.

Example 21

challenge
Show that aΓ—a=0\mathbf{a}\times\mathbf{a}=\mathbf{0} for any vector a=⟨a1,a2,a3⟩\mathbf{a}=\langle a_1,a_2,a_3\rangle.

Example 22

medium
Compute ⟨2,βˆ’1,3βŸ©Γ—βŸ¨0,4,βˆ’2⟩\langle 2, -1, 3 \rangle \times \langle 0, 4, -2 \rangle.

Example 23

easy
Compute ⟨0,0,1βŸ©Γ—βŸ¨1,0,0⟩\langle 0, 0, 1 \rangle \times \langle 1, 0, 0 \rangle.

Example 24

easy
Find ⟨3,0,0βŸ©Γ—βŸ¨0,0,2⟩\langle 3, 0, 0 \rangle \times \langle 0, 0, 2 \rangle.

Example 25

easy
True or false: aΓ—b\mathbf{a}\times\mathbf{b} is perpendicular to a\mathbf{a}.

Example 26

medium
Compute ⟨1,2,3βŸ©Γ—βŸ¨4,5,6⟩\langle 1, 2, 3 \rangle \times \langle 4, 5, 6 \rangle.

Example 27

medium
Find ⟨0,3,βˆ’1βŸ©Γ—βŸ¨2,1,4⟩\langle 0, 3, -1 \rangle \times \langle 2, 1, 4 \rangle.

Example 28

medium
Find the area of the parallelogram with adjacent sides u=⟨1,2,2⟩\mathbf{u}=\langle 1,2,2\rangle and v=⟨3,0,0⟩\mathbf{v}=\langle 3,0,0\rangle.

Example 29

medium
Find the area of the triangle with vertices A=(1,1,0)A=(1,1,0), B=(4,1,0)B=(4,1,0), C=(1,5,0)C=(1,5,0).

Example 30

medium
Find ⟨2,βˆ’1,0βŸ©Γ—βŸ¨0,0,5⟩\langle 2, -1, 0 \rangle \times \langle 0, 0, 5 \rangle.

Example 31

medium
For a=⟨1,1,1⟩\mathbf{a}=\langle 1,1,1\rangle and b=⟨2,2,2⟩\mathbf{b}=\langle 2,2,2\rangle, what is aΓ—b\mathbf{a}\times\mathbf{b}?

Example 32

medium
Find βŸ¨βˆ’1,2,3βŸ©Γ—βŸ¨4,0,βˆ’2⟩\langle -1, 2, 3 \rangle \times \langle 4, 0, -2 \rangle.

Example 33

hard
Find a unit vector perpendicular to both ⟨2,0,1⟩\langle 2, 0, 1 \rangle and ⟨0,3,1⟩\langle 0, 3, 1 \rangle.

Example 34

hard
Find the angle between a=⟨1,0,0⟩\mathbf{a}=\langle 1,0,0\rangle and b=⟨1,1,0⟩\mathbf{b}=\langle 1,1,0\rangle using the magnitude of the cross product.

Example 35

hard
Compute the scalar triple product aβ‹…(bΓ—c)\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c}) for a=⟨1,2,3⟩\mathbf{a}=\langle 1,2,3\rangle, b=⟨0,1,0⟩\mathbf{b}=\langle 0,1,0\rangle, c=⟨0,0,1⟩\mathbf{c}=\langle 0,0,1\rangle.

Example 36

hard
Find the equation of the plane through (1,1,1)(1,1,1) with normal ⟨1,0,1βŸ©Γ—βŸ¨0,1,1⟩\langle 1,0,1\rangle\times\langle 0,1,1\rangle.

Example 37

hard
Compute ⟨3,1,βˆ’2βŸ©Γ—βŸ¨1,βˆ’1,1⟩\langle 3, 1, -2 \rangle \times \langle 1, -1, 1 \rangle.

Example 38

challenge
For a=⟨1,2,2⟩\mathbf{a}=\langle 1,2,2\rangle and b=⟨2,1,βˆ’2⟩\mathbf{b}=\langle 2,1,-2\rangle, find the area of the parallelogram and verify βˆ₯aβˆ₯βˆ₯bβˆ₯sin⁑θ\|\mathbf{a}\|\|\mathbf{b}\|\sin\theta matches.
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Background Knowledge

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

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