Cross product

Multiply two 3D vectors to get a third one perpendicular to both.

Cross product of two 3D vectors gives a third vector perpendicular to both.

Drag the two vectors

|a| = √3²+1²
3.16

|b|
3.16

a × b (z-component) = 8
= signed area of the parallelogram

The widget above shows the 2D cross product — a single number, the z-component, equal to a₁b₂ − a₂b₁ and to the signed area of the parallelogram. The *full* cross product a × b is only defined in 3D, where it's a vector perpendicular to both.

3D cross product

a \times b = (a_{2} b_{3} - a_{3} b_{2}, a_{3} b_{1} - a_{1} b_{3}, a_{1} b_{2} - a_{2} b_{1})

Its magnitude |a||b|sin θ is the parallelogram's area; its direction follows the right-hand rule.

Your turn

Compute the 2D cross product of (3, 0) and (0, 4). What does it represent?

Magnitude

∣ a \times b ∣ = ∣ a ∣∣ b ∣ sin θ

Right-hand rule: curl fingers from a to b, thumb points along a × b.

Vectors Dot product Matrix determinant

Cross product

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