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The cross product

The cross product of two vectors in space is a new vector. It is perpendicular to both, and its length is the area of the parallelogram they span. The coordinates are calculated with a fixed pattern of products. A normal vector of a plane can be found as the cross product of two vectors that lie in the plane. The length, not the vector itself, is the area.

∣u⃗×v⃗∣|\vec u \times \vec v|area of the parallelogram
u⃗×v⃗⊥u⃗\vec u \times \vec v \perp \vec uis perpendicular

Symbols

u⃗, v⃗\vec u,\ \vec vthe vectors in the plane

Example

[1,2,0]×[0,1,3]=[6,−3,1][1, 2, 0] \times [0, 1, 3] = [6, -3, 1]. The area is the length 36+9+1≈6.78\sqrt{36 + 9 + 1} \approx 6.78.

The cross product is a vector. The area is its length, so you must take the square root at the end.
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Part of Mathematics R2: Vectors in space.