Question

Solve the following problems. Show complete solution.

1. Find the volume of the parallel piped with the vertices in the given order: (0, 0, 0), (3, 0, 0), (0, 5, 1), (2, 0 , 5), (3, 5, 1), (5, 0, 5), (2, 5, 6), and (5, 5, 6).

Answer 1

Consider a parallelepiped  the vertices of which have coordinates

$A(0,0,0)B(3,0,0), C(0,5,1), D(2,0,5), \\ A_1(3,5,1),B_1(5,0,5), C_1(2,5,6),D_1(5,5,6)$ .

 Form vectors

$\overline{AB}=(3-0,0-0,0-0)=(3,0,0)\\ \overline{AC}=(0-0,5-0,1-0)=(0,5,1)\\ \overline{AD}=(2-0,0-0,5-0)=(2,0,5)$

$V=\overline{AB}\cdot (\overline{AC}\times \overline{AD})\\ \overline{AC}\times \overline{AD}=(\begin{vmatrix} 5 & 1 \\ 0 & 5 \end{vmatrix},-\begin{vmatrix} 0 & 1 \\ 2 & 5 \end{vmatrix},\begin{vmatrix} 0 & 5 \\ 2 &0 \end{vmatrix})=\\ =(25-0,-(0-2),0-10)=(25,2,-10)\\ V=3\cdot25+0\cdot2+0\cdot(-10)=75$

Answer: $V=75$