Orthogonality, projections and the cross product

A vector normal to a plane

A non-zero vector n is normal to a plane P if it is orthogonal to two non-collinear vectors in P (and therefore to all vectors in P). If n(a, b, c) is normal to P and A(x₀, y₀, z₀) lies in P, a Cartesian equation for P is: a(x – x₀) + b(y – y₀) + c(z – z₀) = 0, or ax + by + cz + d = 0

Orthogonal projection

The orthogonal projection of v onto the line directed by u (where u is non-zero) is the vector: p = ((u·v)/||u||²) · u

This formula is used, in particular, to calculate the distance from a point to a line or a plane.

Cross product (useful reminder)

The cross product u^v of u(x₁, y₁, z₁) and v(x₂, y₂, z₂) is the vector: u^v = (y1.z2 - z1.y2, z1.x2 - x1.z2, x1.y2 - y1.x2)

It is orthogonal to both u and v simultaneously, making it a straightforward tool for finding a normal vector to a plane defined by two direction vectors.

Example: u(1,0,0), v(0,1,0) give u^v = (0·0 − 0·1, 0·0 − 1·0, 1·1 − 0·0) = (0, 0, 1).

Tool	Result	Main use
Scalar product	scalar	angle, orthogonality
Vector product	vector	normal vector, area
Projection	vector	point-to-line distance

Common pitfall

The cross product is defined (in this context) only in three dimensions and is not commutative: u^v = -(v^u). Reversing the order changes the sign of the result, which can distort the orientation of a normal vector.