AHL 3.16 (HL)—Vector product

The vector product creates a perpendicular direction.

a×b is perpendicular to both vectors and has magnitude |a||b|sinθ, making it useful for normals and areas.

Example

(1,0,0)×(0,2,0)=(0,0,2), whose magnitude 2 is the area of the rectangle spanned by the vectors.

Use the determinant pattern, then apply the right-hand rule to interpret orientation and check the magnitude.

Reversing the order changes the sign: a×b=−(b×a); parallel vectors give the zero vector.

For $\mathbf a=(a_1,a_2,a_3)$ and $\mathbf b=(b_1,b_2,b_3)$, compute $\mathbf a\times\mathbf b=(a_2b_3-a_3b_2,\ a_3b_1-a_1b_3,\ a_1b_2-a_2b_1)$. Its magnitude is the area of the spanned parallelogram, so the corresponding triangle area is $\tfrac12|\mathbf a\times\mathbf b|$. The zero vector occurs for non-zero parallel inputs, while order controls the normal's direction.