Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS, where P (-2, 1, 1), Q(-1, 0, 2), R(0, 5, 2), and S(1, 3, 0).
A)19
B)22
C)38
D)80
E)20
F)40
G)44
H)72

Question

Quizplus · Accepted Answer

The volume of a parallelepiped with adjacent edges PQ, PR, and PS is given by the scalar triple product of the three vectors PQ, PR, and PS. That is,$$V = \mathbf{PQ} \cdot (\mathbf{PR} 	imes \mathbf{PS})$$where$$\mathbf{PQ} = \langle -1 - (-2), 0 - 1, 2 - 1 angle = \langle 1, -1, 1 angle$$$$\mathbf{PR} = \langle 0 - (-1), 5 - 0, 2 - 2 angle = \langle 1, 5, 0 angle$$$$\mathbf{PS} = \langle 1 - (-2), 3 - 1, 0 - 1 angle = \langle 3, 2, -1 angle$$and$$\mathbf{PR} 	imes \mathbf{PS} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 5 & 0 \\ 3 & 2 & -1 \end{vmatrix} = \langle -2, 1, 11 angle$$Therefore,$$V = \langle 1, -1, 1 angle \cdot \langle -2, 1, 11 angle = (1)(-2) + (-1)(1) + (1)(11) = 19$$So, the volume of the parallelepiped is 19 cubic units.

Find the Volume of the Parallelepiped with Adjacent Edges PQ