Question

Volume of the parallelepiped having vertices at $O\equiv (0,0,0),A\equiv (2,–2,1),B\equiv (5,–4,4)$and $c\equiv (1,–2,4)$ is

• A. $5$ $cuunits$
• B. $10$ $cuunits$
• C. $15$ $cuunits$
• D. $20$ $cuunits$

Answer 1

The correct option is B

$10$ $cuunits$

Explanation for correct option

Volume of parallelepiped:
Given vertices $O\equiv (0,0,0),A\equiv (2,–2,1),B\equiv (5,–4,4)$and $c\equiv (1,–2,4)$
$$\begin{gathered} \overrightarrow{OA}=(2-0)\hat{i}+(-2-0)\hat{j}+(1-0)\hat{k} \\ =2\hat{i}-2\hat{j}+\hat{k} \end{gathered}$$
Similarly,
$\overrightarrow{OB}=5\hat{i}-4\hat{j}+4\hat{k}\&\overrightarrow{OC}=\hat{i}-2\hat{j}+4\hat{k}$
Volume of parallelepiped $\mathbf{=}\left[\overrightarrow{\mathrm{OA}}\overrightarrow{\mathrm{OB}}\overrightarrow{\mathrm{OC}}\right]$
$$\begin{gathered} =\left|\begin{array}{ccc}2& -2& 1\\ 5& -4& 4\\ 1& -2& 4\end{array}\right| \\ =2(-16+8)+2(20-4)+1(-10+4) \\ =-16+32-6 \\ =10\mathrm{cu}\mathrm{units} \end{gathered}$$

Hence option(B) is correct.