Question

(1 point) find the volume of the parallelepiped with one vertex at (5,−5,−1),(5,−5,−1), and adjacent vertices at (−1,−12,4),(−1,−12,4), (11,−9,−3),(11,−9,−3), and (0,2,−3).(0,2,−3).

Answer 1

The volume is the scalar triple product of the direction vectors from the first point to the others. That is computed as the magnitude of the determinant of the matrix of vector values.

a = (-1, -12, 4) - (5, -5, -1) = (-6, -7, 5)

b = (11, -9, -3) - (5, -5, -1) = (6, -4, -2)

c = (0, 2, -3) - (5, -5, -1) = (-5, 7, -2)

Then |(a×b)•c| is

$\left|det\left[\begin{array}{ccc}-5&7&-2\\-6&-7&5\\6&-4&-2\end{array}\right]\right|=|-176|=176$

The volume is 176 cubic units.