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Planes and normal vectors

A plane has an equation ax+by+cz=dax + by + cz = d. The vector of the coefficients is a normal vector, and it is perpendicular to every vector in the plane. A point lies in the plane when its coordinates satisfy the equation. Two planes are perpendicular when their normal vectors have dot product zero. You therefore read the normal vector straight from the equation, without first calculating a cross product.

ax+by+cz=dax + by + cz = dthe equation of the plane
n⃗=[a,b,c]\vec n = [a, b, c]normal vector

Symbols

n⃗\vec nthe normal vector
ddthe constant

Example

The plane 2x−y+3z=52x - y + 3z = 5 has normal vector [2,−1,3][2, -1, 3]. The point (1,2,1)(1, 2, 1) gives 2−2+3=32 - 2 + 3 = 3, so it does not lie in the plane. (1,1,1)(1, 1, 1) in x+y−z=1x + y - z = 1 gives 11.

The constant on the right-hand side is not a fourth coordinate of the normal vector.
Practise vectors in space for free →

← Length and dot product in space · The cross product →

Part of Mathematics R2: Vectors in space.