1. We can use vectors and the cross product to construct planes quite easily.
2. Determining a Plane From a Point and Normal Vector. Given a vector
and a point , there exists a unique plane which is perpindicular to and contains .
How? Well, let be an arbitrary point on the plane. Then the vector would be parallel to the plane. Being parallel to the plane implies its orthogonal to :
This gives us an equation
So any point lies on the plane if it satisfies this equation.
3. Determine a Plane from Three Points. Given three points , , , find a plane containing these points.
We construct the vectors and . Take the cross product, which produces the normal vector
If we write and , then
describes the plane. (It follows from our last construction of the plane.)
Exercise 1 [Math.SE]. Determine the plane containing point and going through the intersection line of the planes and .
Exercise 2 [Math.SE]. Let be the line given by , , . Then intersects the plane at the point . Find parametric equations for the line through which lies on plane and is perpendicular to .
Exercise 3 [Math.SE]. Find normal form equation of plane containing and Line: where is some parameter.