If \(\mathbf{i}+\mathbf{j}-\mathbf{k}\) and \(2 \mathbf{i}-3 \mathbf{j}+\mathbf{k}\) are adjacent sides of a parallelogram, then the lengths of its diagonals are

\(\begin{array}{ll}\text { Let } & \mathbf{A B}=\mathbf{i}+\mathbf{j}-\mathbf{k} \\ \text { and } & \mathbf{B C}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k}\end{array}\)



The diagonal \(\mathbf{A C}=\mathbf{A B}+\mathbf{B C}\)
\[
\begin{aligned}
&=(\mathbf{i}+\mathbf{j}-\mathbf{k})+(2 \mathbf{i}-3 \mathbf{j}+\mathbf{k}) \\
&=3 \mathbf{i}-2 \mathbf{j}
\end{aligned}
\]
The length of diagonal
\[
\mathbf{A C}=\sqrt{(3)^{2}+(-2)^{2}}=\sqrt{9+4}=\sqrt{13}
\]
The diagonal \(\mathbf{D B}=\mathbf{A B}-\mathbf{A D}\)
\[
\begin{aligned}
&=(\mathbf{i}+\mathbf{j}-\mathbf{k})-(2 \mathbf{i}-3 \mathbf{j}+\mathbf{k}) \\
&=-\mathbf{i}+4 \mathbf{j}-2 \mathbf{k}
\end{aligned}
\]
The length of diagonal
\[
\text { DB }=\sqrt{1+16+4}=\sqrt{21}
\]