Here's an outline of a proof, I'm not sure where you will want to begin.
- Show that the magnitude of the cross product of two vectors (a and b) gives the area of the parallelogram they define. Thus, the area of the base of the tetrahedron is half of this.
- The direction of the cross product is perpendicular to the plane containing them, so it is in the direction of the height of the tetrahedron. Show that the dot product of vector c with the unit vector of height will give the height of the prism from the base defined by a and b.
- Given that the volume of any pyramid is one third the area of the base times the height, you can put this all together to finish the proof.
Hope this helps,