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 \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.

Read the full concept explanation →

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 produces a vector perpendicular to both inputs, with magnitude equal to the area of the parallelogram they span. It is anti-commutative: \mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a}).

Common stuck point: The cross product is only defined for 3D vectors (and 7D, but that is rarely encountered). Also, order matters: \mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a}).

Sense of Study hint: Cover one row at a time in the 3x3 determinant setup and compute the 2x2 determinant of what remains.

Worked Examples

Example 1

medium
Find \langle 1, 0, 0 \rangle \times \langle 0, 1, 0 \rangle.

Solution

  1. 1
    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.
  2. 2
    Step 2: = \langle 0 \cdot 0 - 0 \cdot 1, 0 \cdot 0 - 1 \cdot 0, 1 \cdot 1 - 0 \cdot 0 \rangle.
  3. 3
    Step 3: = \langle 0, 0, 1 \rangle.
  4. 4
    Check: \hat{\mathbf{i}} \times \hat{\mathbf{j}} = \hat{\mathbf{k}} by the right-hand rule ✓

Answer

\langle 0, 0, 1 \rangle
The cross product of two vectors gives a third vector perpendicular to both. For the standard basis vectors, \hat{\mathbf{i}} \times \hat{\mathbf{j}} = \hat{\mathbf{k}} follows the right-hand rule.

Example 2

hard
Find \langle 2, 3, 1 \rangle \times \langle 1, -1, 2 \rangle.

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
Is \mathbf{a} \times \mathbf{b} = \mathbf{b} \times \mathbf{a}?
← Back to Cross Product Formula Explained Practice Mode

Background Knowledge

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

dot productvector operationsdeterminant