If A = 2i - 2j + k, B = i + 2j + 2k, and C = - i + 2j + k, the area of a parallelepiped whose edges are defined by the vectors A, B, and C is
A) 3.
B) 6.
C) 8.
D) 9.

Question

Quizplus · Accepted Answer

The volume of a parallelepiped defined by vectors A, B, and C is given by the absolute value of the scalar triple product of A, B, and C, which is |A · (B × C)|. First, find the cross product B × C:B × C = |i  j  k |        |1  2  2|        |-1 2  1|= i(2*1 - 2*2) - j(1*1 - 2*-1) + k(1*2 - 2*-1)= i(2 - 4) - j(1 + 2) + k(2 + 2)= -2i - 3j + 4kThen, find the dot product A · (B × C):A · (B × C) = (2i - 2j + k) · (-2i - 3j + 4k)= 2*(-2) + (-2)*(-3) + 1*4= -4 + 6 + 4= 6The volume of the parallelepiped, which is also the area in this context, is the absolute value of this result, which is |6| = 6.

If a = 2i - 2j + K, B =