The volume of the parallelepiped determined by the vectors a, b, and c is the magnitude of their scalar triple product: V = |a . (b x c)|.
Applications:
1. Use the scalar triple product to show that the vectors a = (1,4,-7), b = (2,-1,4), and c = (0,-9,18) are coplanar; that is, they lie in the same plane.
2. Find the volume of the parallelepiped determined by the vectors a = (1,0,6), b = (2,3,-8), and c = (8,-5,6).
3. Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS, if P(1,1,1), Q(2,0,3), R(4,1,7), and S(3,-1,-2).