If \(2 \mathbf{i}+3 \mathbf{j}, \mathbf{i}+\mathbf{j}+\mathbf{k}\) and \(\lambda \mathbf{i}+4 \mathbf{j}+2 \mathbf{k}\) taken in an order are coterminous edges of a parallelopiped of volume \(2 \mathrm{cu}\) units, then value of \(\lambda\) is

Given, let \(A=2 \mathbf{i}+3 \mathbf{j}+0 \mathbf{k}\)
\(B=\mathbf{i}+\mathbf{j}+\mathbf{k}\)
\(C=\lambda i+4 j+2 k\)
If \(A, B, C\) are the coterminous edges of a parallelopiped then its volume is
\[
=(A \times B) \cdot C
\]
\[
=\{(2 \mathbf{i}+3 \mathbf{j}+0 \mathbf{k}) \times(\mathbf{i}+\mathbf{j}+\mathbf{k})\}
\]
\((\lambda \mathbf{i}+4 \mathbf{j}+2 \mathbf{k})\)
\(=(3 \mathbf{i}-2 \mathbf{j}-\mathbf{k}) \cdot(\lambda \mathbf{i}+4 \mathbf{j}+2 \mathbf{k})\)
\(=3 \lambda-8-2\)
\(=3 \lambda-10\)
Given volume \(=2\)
\[
\begin{aligned}
& \Rightarrow & 3 \lambda-10 &=2 \\
\Rightarrow & & 3 \lambda &=12 \\
\Rightarrow & \lambda &=4
\end{aligned}
\]