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⟩ and b=⟨b1,b2,b3⟩ is a new vector a×b that is perpendicular to both a and b. Its magnitude equals the area of the parallelogram formed by a and 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.

Showing a random 20 of 50 problems.

Example 1

medium
Compute ⟨2,3,4⟩×⟨5,6,7⟩.

Example 2

easy
What does the magnitude of a×b represent geometrically?

Example 3

easy
Is a×b=b×a?

Example 4

medium
Compute ⟨2,−1,3⟩×⟨0,4,−2⟩.

Example 5

challenge
For a=⟨1,2,2⟩ and b=⟨2,1,−2⟩, find the area of the parallelogram and verify ∥a∥∥b∥sin⁡θ matches.

Example 6

easy
True or false: a×b is perpendicular to a.

Example 7

easy
Is ⟨0,0,5⟩ perpendicular to both ⟨1,0,0⟩ and ⟨0,1,0⟩?

Example 8

hard
Compute ⟨3,1,−2⟩×⟨1,−1,1⟩.

Example 9

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

Example 10

medium
If a×b=⟨4,−2,1⟩, what is b×a?

Example 11

medium
Find ⟨2,−1,0⟩×⟨0,0,5⟩.

Example 12

hard
Compute the scalar triple product a⋅(b×c) for a=⟨1,2,3⟩, b=⟨0,1,0⟩, c=⟨0,0,1⟩.

Example 13

medium
Two vectors have magnitudes 3 and 5 with a 30∘ angle between them. Find ∥a×b∥.

Example 14

medium
Find ⟨1,0,0⟩×⟨0,1,0⟩.

Example 15

hard
Find ⟨2,3,1⟩×⟨1,−1,2⟩.

Example 16

easy
Compute ⟨1,2,3⟩×⟨1,2,3⟩.

Example 17

easy
Compute ⟨0,0,1⟩×⟨1,0,0⟩.

Example 18

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

Example 19

medium
For a=⟨1,1,1⟩ and b=⟨2,2,2⟩, what is a×b?

Example 20

easy
Compute ⟨1,0,0⟩×⟨0,1,0⟩.
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