Find the volume of the parallelepiped with the given vertices. $\mathrm { A } ( 0,0,0 ) , \mathrm { B } ( 5,5,0 ) , \mathrm { C } ( 5,0,6 ) , \mathrm { D } ( 0,5,5 ) , \mathrm { E } ( 6,5,6 ) , \mathrm { F } ( 5,5,7 ) , \mathrm { G } ( 5,6,5 ) , \mathrm { H } ( 6,6,7 )$
A) $276 \text { cubic units }$
B) $275 \text { cubic units }$
C) $277 \text { cubic units }$
D) $279 \text { cubic units }$
E) $278 \text { cubic units }$

Question

Find the volume of the parallelepiped with the given vertices.  $\mathrm { A } ( 0,0,0 ) , \mathrm { B } ( 5,5,0 ) , \mathrm { C } ( 5,0,6 ) , \mathrm { D } ( 0,5,5 ) , \mathrm { E } ( 6,5,6 ) , \mathrm { F } ( 5,5,7 ) , \mathrm { G } ( 5,6,5 ) , \mathrm { H } ( 6,6,7 )$ 
A) $276 \text { cubic units }$
B) $275 \text { cubic units }$
C) $277 \text { cubic units }$
D) $279 \text { cubic units }$
E) $278 \text { cubic units }$

Quizplus · Accepted Answer

The volume of a parallelepiped can be found using the scalar triple product of vectors. The vectors can be formed using the given vertices. For instance, AB, AC, and AD can be used to form the necessary vectors. The coordinates of these vectors are AB = (5, 5, 0), AC = (5, 0, 6), and AD = (0, 5, 5). The scalar triple product is |AB · (AC × AD)|, which gives the volume. Calculating this gives a volume of 275 cubic units.

Find the Volume of the Parallelepiped with the Given Vertices