How to Find the Cross Product of Two Vectors

Calculates the cross product of two vectors in space. It is perpendicular to both, and its magnitude is the area of the parallelogram they span. Half of that is the area of the triangle.

Two vectors in space give a third one perpendicular to both. That is the cross product.

\vec{a} \times \vec{b} = (y_1z_2 - z_1y_2,\ z_1x_2 - x_1z_2,\ x_1y_2 - y_1x_2)

Each component multiplies the other two components crosswise and subtracts. The magnitude of the cross product is the area of the parallelogram the two vectors span, and half of that is the area of the triangle.

How it differs from the dot product

The dot product returns a number; the cross product returns a vector. The dot product measures how far two vectors point the same way, the cross product how far they point differently. Parallel vectors have a cross product of zero and span no area at all.

Example

For $(2, 3, 4)$ and $(5, 6, 7)$ , the $x$ component is $3 \times 7 - 4 \times 6 = -3$ , the $y$ component is $4 \times 5 - 2 \times 7 = 6$ and the $z$ component is $2 \times 6 - 3 \times 5 = -3$ . The magnitude is $\sqrt{9 + 36 + 9} = \sqrt{54}$ , about 7.348, and the triangle is half that, about 3.674.

Notes

Order matters: $\vec{b} \times \vec{a}$ is $\vec{a} \times \vec{b}$ with every sign flipped. The dot product does not behave this way.

For vectors in a plane, set $z$ to 0. Only the $z$ component of the cross product survives, and its value is the area of the parallelogram.