Question

If $A,B,C,D$ are $(1,1,1),(2,1,3),(3,2,2),(3,3,4)$ respectively, then find the volume of the parallelopiped (in cubic units) with $AB,AC$ and $AD$ as the concurrent edges.

Answer 1

Given that $A,B,C$ and $D$ are $(1,1,1),(2,1,3),(3,2,2)$ and $(3,3,4)$ respectively.
We need to find the volume of the parallelopiped with $AB,AC$ and $AD$ as the concurrent edges.
The volume of the parallelopiped whose edges are $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ is $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]=\overrightarrow{a}.(\overrightarrow{b}\times \overrightarrow{c}).$
$\overrightarrow{AB}=(2-1)\hat{i}+(1-1)\hat{j}+(3-1)\hat{k}$
$=\hat{i}+2\hat{k}$
$\overrightarrow{AC}=(3-1)\hat{i}+(2-1)\hat{j}+(2-1)\hat{k}$
$=2\hat{i}+\hat{j}+\hat{k}$
$\overrightarrow{AB}=(3-1)\hat{i}+(3-1)\hat{j}+(4-1)\hat{k}$
$=2\hat{i}+2\hat{j}+3\hat{k}$
$\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]=\left|\begin{array}{ccc}1& 0& 2\\ 2& 1& 1\\ 2& 2& 3\end{array}\right|$

$=1(3-2)-0+2(4-2)$
$=1+4$
$=5$ cubic units