The volume of a parallelopiped whose edges are represented by $-12 \hat{i} + \lambda \hat{k}$, $3 \hat{j} - \hat{k}$ and $2 \hat{i} + \hat{j} - 15 \hat{k}$ is $546$. Find the value of $\lambda.$
Let $\overrightarrow{a} = -12 \hat{i} + \lambda \hat{k}$, $\;$ $\overrightarrow{b} = 3 \hat{j} - \hat{k}$ $\;$ and $\;$
$\overrightarrow{c} = 2 \hat{i} + \hat{j} - 15 \hat{k}$
$\begin{aligned}
\text{Volume of parallelopiped} = \left[\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}\right] & = \overrightarrow{a} \cdot \left(\overrightarrow{b} \times \overrightarrow{c}\right) \\\\
& = \begin{vmatrix}
-12 & 0 & \lambda \\
0 & 3 & -1 \\
2 & 1 & -15
\end{vmatrix} \\\\
& = - 12 \times \left(-45 + 1\right) + 0 + \lambda \times \left(0 - 6\right) \\\\
& = 528 - 6 \lambda
\end{aligned}$
Given: $\;$ volume of parallelopiped $= 546$
i.e. $\;$ $528 - 6 \lambda = 546$ $\implies$ $- 6 \lambda = 18$ $\implies$ $\lambda = -3$