Q69 / 170 · Manipal (MET) 2020 · Mathematics
Three concurrent edges of a parallelopiped are given by
$\begin{aligned} \mathbf{a} & =2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \\ \mathbf{b} & =\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}} \\ \text { and } \quad \mathbf{c} & =2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}\end{aligned}$
The volume of the parallelopiped is
- Correct answerA14 cu units
- B20 cu units
- C25 cu units
- D60 cu units
Solution
Given edges of a parallelopiped are
$\mathbf{a}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and $\mathbf{c}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$
$\therefore$ Volume of parallelopiped $=[\mathbf{a b c}]$
$=\left\|\begin{array}{ccc}2 & -3 & 1 \\ 1 & -1 & 2 \\ 2 & 1 & -1\end{array}\right\|$
$=|2(1-2)+3(-1-4)+1(1+2)|$
$=|-2-15+3|=14$ cu units
Hence, the correct answer is option: 'A'.