Find the volume of the parallelepiped with the given vertices. $\mathrm { A } ( 0,0,0 ) , \mathrm { B } ( 6,0,0 ) , \mathrm { C } ( 6 , - 4,5 ) , \mathrm { D } ( 0 , - 4,5 ) , \mathrm { E } ( 6,7,5 ) , \mathrm { F } ( 0,7,5 ) , \mathrm { G } ( 0,7,8 ) , \mathrm { H } ( 6,5,8 )$
A) $\text { Volume } = 331 \text { cubic units }$
B) $\text { Volume } = 332 \text { cubic units }$
C) $\text { Volume } = 329 \text { cubic units }$
D) $\text { Volume } = 330 \text { cubic units }$
E) $\text { Volume } = 328 \text { cubic units }$

Question

Find the volume of the parallelepiped with the given vertices.  $\mathrm { A } ( 0,0,0 ) , \mathrm { B } ( 6,0,0 ) , \mathrm { C } ( 6 , - 4,5 ) , \mathrm { D } ( 0 , - 4,5 ) , \mathrm { E } ( 6,7,5 ) , \mathrm { F } ( 0,7,5 ) , \mathrm { G } ( 0,7,8 ) , \mathrm { H } ( 6,5,8 )$ 
A) $\text { Volume } = 331 \text { cubic units }$
B) $\text { Volume } = 332 \text { cubic units }$
C) $\text { Volume } = 329 \text { cubic units }$
D) $\text { Volume } = 330 \text { cubic units }$
E) $\text { Volume } = 328 \text { cubic units }$

Quizplus · Accepted Answer

The volume of a parallelepiped can be found using the scalar triple product of vectors. Here, we can use vectors AB, AD, and AE as the edges of the parallelepiped. The coordinates of these vectors are AB = (6, 0, 0), AD = (0, -4, 5), and AE = (6, 7, 5). The scalar triple product is |AB · (AD × AE)|, which gives the volume. Calculating the cross product of AD and AE and then dotting it with AB yields the volume = 330 cubic units.

Find the Volume of the Parallelepiped with the Given Vertices