A plane in
three-space can be defined by one
point in the plane and a
normal vector orthogonal to the plane.
Vector Equation of a Plane:
Given a point P0(x0, y0, z0) in the plane and a normal vector n, let P(x, y, z) be an arbitrary point in the plane. Let r and r0 be the position vectors of P and P0 respectively. Subtracting r from r0 gives us a vector inside the plane, which is orthogonal to n. Thus:
n . (r - r0) = 0
or
n . r = n . r0
Scalar Equation of a Plane:
Given a point P0(x0, y0, z0) in the plane and a normal vector n = , let P(x, y, z) be an arbitrary point in the plane. The vector equation then becomes:
a(x-x0) + b(y-y0) + c(z-z0) = 0
Linear Equation of a Plane:
ax + by + cz = d
where d = ax0 + by0 + cz0
This node made possible by Calculus Concepts and Contexts by James Stewart.