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
Consider two planes with Cartesian equations,
Determine the angle between the planes.
For the two planes we have normals and . Therefore, the angle between the planes is
Consider four planes with the Cartesian equations:
- Establish that two of them are parallel and the other two are perpendicular.
- For the two parallel planes, find the distance between them.
- The second and fourth planes are parallel since they have the same normal vector, .
To see that the first and third are perpendicular, we denote the normal vectors by and , respectively, and compute
Hence, they are perpendicular.
- Denoting the second and fourth planes by and , respectively we have Cartesian equations
Therefore, the distance between them is