Find an equation and a parametric representation for the plane passing through the points (1,0,1) , (2,3,1) , (5,4,5).

The difference between two points in a plane is a vector that also lies in the plane. Therefore, we can use the points:

to determine two vectors that lie in the plane by subtracting the first point for the other two.

Choosing the first point as Xo, we create the parameterized plane using the vectors as follows:

The perpendicular vector n can be found by taking the cross product of the two vectors in the plane:

The normal vector is dotted with a point in the plane to find d:

The normal vector is now dotted with (x,y,z) to obtain the equation in standard form: