Prove that if and are two planes which are not parallel then they intersect in a line.
Proof. Let the Cartesian equations of and be given by
respectively. Then, the intersection is given by the common solutions of these two equations. Since and are not parallel, we know they do not have the same normal vector so that for all . Further, since the normals are nonzero, we know each equation has at least one nonzero coefficient. Without loss of generality, let . Then,
Substituting into the Cartesian equation for we have
is the set of solutions for the points on . But, we know at least one of or is nonzero, otherwise . Hence, we have the equation for a line. Therefore, is a line
Consider a plane given by the equation
Find the Cartesian equation for a plane parallel to this one and the same distance as this plane from the point .
Since the requested plane is parallel to the given plane we know that they must have the same normal vector, . Therefore, the Cartesian equation of the requested plane is of the form
From the previous exercise (Section 13.17, Exercise #19) we know the distance from to a plane is given by the formula
Therefore, the distance from the given plane to the point is
Since the distance from the point to the requested plane must be the same we must have
(Since the solution belongs to the other plane.)
Consider the tetrahedron with vertices at the origin and at the points where the plane
intersects the coordinate axes. Compute the volume of this tetrahedron.
First, the intercepts of the plane are given by . Then from a previous exercise (Section 13.14, Exercise #13) we know that the volume of a tetrahedron with vertices is
Letting we have