Find the volume of the parallelepiped with the following vertices. $$( 0,0,0 ) , ( 5,0,0 ) , ( 0,9,3 ) , ( 5,9,3 ) , ( 2,0,4 ) , ( 7,0,4 ) , ( 2,9,7 ) , ( 7,9,7 )$$
A) 180
B) 167
C) 60
D) 30
E) 22

Question

Find  the volume of the parallelepiped with the following vertices. $$( 0,0,0 ) , ( 5,0,0 ) , ( 0,9,3 ) , ( 5,9,3 ) , ( 2,0,4 ) , ( 7,0,4 ) , ( 2,9,7 ) , ( 7,9,7 )$$
A) 180
B) 167
C) 60
D) 30
E) 22

Quizplus · Accepted Answer

The volume of a parallelepiped can be found using the scalar triple product of vectors. The vectors can be formed using three adjacent vertices from the origin, for example, using vertices (0,0,0), (5,0,0), (0,9,3), and (2,0,4), we can form vectors $\vec{a} = (5,0,0), \vec{b} = (0,9,3), \text{ and } \vec{c} = (2,0,4)$. The volume is the absolute value of the scalar triple product of these vectors, which is $|\vec{a} \cdot (\vec{b} \times \vec{c})|$. Calculating $\vec{b} \times \vec{c}$ gives a vector, and then $\vec{a}$ dotted with this result gives a scalar. For these vectors, the calculation yields a volume of 180 cubic units.

Find the Volume of the Parallelepiped with the Following (0,0,0),(5,0,0),(0,9,3),(5,9,3),(2,0,4),(7,0,4),(2,9,7),(7,9,7)( 0,0,0 ) , ( 5,0,0 ) , ( 0,9,3 ) , ( 5,9,3 ) , ( 2,0,4 ) , ( 7,0,4 ) , ( 2,9,7 ) , ( 7,9,7 )(0,0,0),(5,0,0),(0,9,3),(5,9,3),(2,0,4),(7,0,4),(2,9,7),(7,9,7)

Find the Volume of the Parallelepiped with the Following $( 0,0,0 ) , ( 5,0,0 ) , ( 0,9,3 ) , ( 5,9,3 ) , ( 2,0,4 ) , ( 7,0,4 ) , ( 2,9,7 ) , ( 7,9,7 )$