Definition

Cross product

A \times B = ∣ A ∣∣ B ∣ sin θ n

Statement

A vector perpendicular to both A and B, with magnitude equal to the area of the parallelogram they span; direction set by the right-hand rule.

Symbols and units

$A, B$	vectors being multiplied	—
$A \times B$	cross product	—
$θ$	angle between the vectors	degrees
$n$	unit vector perpendicular to both	—

When it applies

Defined only in three dimensions (or treated as the z-component in a 2-D plane).

Don't use when

Not defined the same way outside three dimensions; in 2-D problems only its z-component survives and is often treated as a scalar.

Derivation

Magnitude equals the area of the parallelogram with sides A and B:

∣ A \times B ∣ = ∣ A ∣∣ B ∣ sin θ

Direction is perpendicular to both, fixed by the right-hand rule: curl the fingers from A to B, the thumb gives the direction.

In components:

A \times B = (A_{y} B_{z} - A_{z} B_{y})  + (A_{z} B_{x} - A_{x} B_{z})  + (A_{x} B_{y} - A_{y} B_{x}) k

Worked examples

Area of a rectangle from a cross product

A = (2, 0, 0), B = (0, 3, 0):

A \times B = (0, 0, 6)

Magnitude 6 equals the rectangle's area 2 × 3; the direction +z follows from curling the fingers from A to B.

Watch out for

The cross product is anticommutative: A × B = −B × A. Order matters, unlike the dot product.

Connects to