A plane in 3D space can be described by a vector equation involving a fixed point and two direction vectors lying in the plane. Planes are often denoted by Π (capital pi).

If the position vector of the fixed point is a, and two non-parallel direction vectors in the plane are b and c, then the plane is represented by:

Here, λ and μ are parameters that can take any real values, allowing r to move freely across the entire surface of the plane.

Example

Find the vector equation of the plane through points A(1,2,3), B(2,1,4), and C(3,2,1).

First, form direction vectors from one point, e.g., A, to the others:

Thus, a suitable vector equation is: