Practice Cross 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 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.

Showing a random 20 of 50 problems.

Example 1

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

Example 2

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

Example 3

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

Example 4

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

Example 5

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.

Example 6

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

Example 7

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 8

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

Example 9

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 10

medium
If aΓ—b=⟨4,βˆ’2,1⟩\mathbf{a}\times\mathbf{b}=\langle 4,-2,1\rangle, what is bΓ—a\mathbf{b}\times\mathbf{a}?

Example 11

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

Example 12

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 13

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 14

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

Example 15

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

Example 16

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

Example 17

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

Example 18

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 19

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 20

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