Two adjacent sides of a parallelogram are \(2 \hat{i}-4 \hat{j}+5 \hat{k}\) and \(\hat{i}-2 \hat{j}-3 \hat{k}\), then the unit vector parallel to its diagonal is

Let \(\vec{a}\) and \(\vec{b}\) be the adjacent sides of a parallelogram, where
\(\begin{aligned}
& \overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}+5 \hat{\mathrm{k}} \\
& \overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}
\end{aligned}\)
Let diagonal be \(\overrightarrow{\mathrm{c}}\)
\(\begin{aligned}
\vec{c} & =\vec{a}+\vec{b} \\
\vec{c} & =2 \hat{i}-4 \hat{j}+5 \hat{k}+\hat{i}-2 \hat{j}-3 \hat{k} \\
& =3 \hat{i}-6 \hat{j}+2 \hat{k}
\end{aligned}\)
\(\text { Magnitude of } \begin{aligned}
\vec{c} & =\sqrt{3^2+(-6)^2+(2)^2} \\
& =\sqrt{49}=7
\end{aligned}\)
\(\therefore \quad\) Unit vector in direction of diagonal \(\overrightarrow{\mathrm{c}}\) is
\(\begin{aligned}
& =\frac{\overrightarrow{\mathrm{c}}}{|\overrightarrow{\mathrm{c}}|} \\
& =\frac{1}{7}(3 \hat{\mathrm{i}}-6 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}) \\
& =\frac{3}{7} \hat{\mathrm{i}}-\frac{6}{7} \hat{\mathrm{j}}+\frac{2}{7} \hat{\mathrm{k}}
\end{aligned}\)