Thread: Smaller tetrahedron possible limited by a plane

I need help with this:
Find the equation of a plane that contains the point (1, 2, 1) and intersects the coordinate axes forming the smaller (least volume) tetrahedron possible.

I really have no idea how to proceed.

Originally Posted by nvwxgn

I need help with this:
Find the equation of a plane that contains the point (1, 2, 1) and intersects the coordinate axes forming the smaller (least volume) tetrahedron possible.

I really have no idea how to proceed.

Here is an outline of what needs to be done.

Lets start with what we know


Using this we can make a generic equation of of this plane note that


and


This gives the equation of the plane as

Note also since must be in the plane we get



(we will need this constraint later)

Note the volume of the tetrahedron is given by



Now we need to minimize this using Lagrange multipliers. This gives the system



Solving I get

This gives a volume of .