Question

Volume of the parallelopiped having vertices at $O\equiv (0,0,0),A\equiv (2,-2,1),B\equiv (5,-4,4),$ and $C=(1,-2,4)$ is

• A. $5$cu unit
• B. $10$ cu unit
• C. $15$cu unit
• D. $20$cu unit

Answer 1

The correct option is B $10$ cu unit

Given, $\overrightarrow{OA}=2\hat{i}-2\hat{j}-\hat{k}$
$\overrightarrow{OB}=5\hat{i}-4\hat{j}+4\hat{k}$ and $\overrightarrow{OC}=\hat{i}-2\hat{j}+4\hat{k}$
Volume of parallelopiped $=\left[\overrightarrow{OA}\overrightarrow{OB}\overrightarrow{OC}\right]$
$=\left|\begin{array}{ccc}2& -2& 1\\ 5& -4& 4\\ 1& -2& 4\end{array}\right|$
$=2(-16+8)+2(20-4)+1(-10+4)$
$=10$ cu unit.