Planes

Equation of planes can expressed in either vector or cartesian form. Vector equation is the one containing only vectors. Cartesian equation is in the form: .

Contains a point and parallel to 2 vectors

Suppose a plane:

• is parallel to both and where
• contains

Equation for the plane is:

Contains a point and normal is given

Suppose a plane:

• contains
• has a normal

Equation for the plane is:

Contains 3 points

Suppose a plane contains ( are the position vectors of respectively).

Normal to a plane

Suppose is a plane. is a normal to the plane.

Angle between 2 planes

Consider the two planes:

The angle between the planes is given by:

Here are normal to the planes .

Shortest distance from a point

Consider the plane .

• is a normal to the plane
• is the position vector of any known point on the plane
• is the position vector to the arbitrary point

Intersection

In 3D, to prove 2 planes intersect, it has to be proven that there is a point satisfiying both of the planes.

Of 2 planes

Can either be a:

• Plane - when the planes coincicde
• Line - otherwise

Equation of the line of intersection can be found by:

• Solving with respect to
• Subject and symmetric form can be found

Of 3 planes

Can either be a:

• Plane - when the planes coincide
• Line - when the lines of intersection between the planes pairwise coincide
• Point - otherwise

First pairwise intersection of the planes must be found. And then intersection of those 2 can be found.