Chapter 11, Problem 14CT

To determine

Find the volume of the parallelepiped with the given vertices.

Answer to Problem 14CT

  $\boxed{200}$

Explanation of Solution

Given information:

Find the volume of the parallelepiped with the given vertices.

  $\begin{array}{l}A\left(0,0,0\right),B\left(2,0,0\right),C\left(2,4,0\right),D\left(0,4,0\right)\\ E\left(0,0,6\right),F\left(2,0,6\right),G\left(2,4,6\right),H\left(0,4,6\right)\end{array}$

  

Calculation:

Consider the vertices of the parallelepiped,

  $\begin{array}{l}A\left(0,0,5\right),B\left(0,10,5\right),C\left(4,10,5\right),D\left(4,0,5\right)\\ E\left(0,1,0\right),F\left(0,11,0\right),G\left(4,11,0\right),H\left(4,1,0\right)\end{array}$

To find the volume compute following vectors,

  $\begin{array}{l}\overline{AB}=\left(0-0,10-0,5-5\right)\\ =\left(0,10,0\right)\\ \overline{AD}=\left(4-0,0-0,5-5\right)\\ =\left(4,0,0\right)\\ \overline{AE}=\left(0-0,1-0,0-5\right)\\ =\left(0,1,-5\right)\end{array}$

Vector $\overline{AB},\overline{AD},\overline{AE}$ represent the length, width and height of the parallelepiped the determinant of these three vector will give the volume of the parallelepiped.

  $\begin{array}{l}\left(\overline{AB}\times \overline{AD}\right)\cdot \overline{AE}\left|\begin{array}{ccc}0& 10& 0\\ 4& 0& 0\\ 0& 1& -5\end{array}\right|\\ =0\left[0\times \left(-5\right)-0\left(1\right)\right]-10\left[4\times \left(-5\right)-0\times 0\right]+0\left[4\times \left(1\right)-0\times \left(0\right)\right]\\ =-10\left(-20\right)\end{array}$

  $=200$

Hence the volume of the parallelepiped is $\boxed{200}$ .