If $\overrightarrow{P}=3\hat{\mathrm{i}}+\sqrt{3}\hat{\mathrm{j}}+2\hat{\mathrm{k}}$ and $\overrightarrow{Q}=4\hat{\mathrm{i}}+\sqrt{3}\hat{\mathrm{j}}+2.5\hat{\mathrm{k}}$  then, the unit vector in the direction of $\overrightarrow{P}\times \overrightarrow{Q}$ is $\frac{1}{x}(\sqrt{3}\hat{\mathrm{i}}+\hat{\mathrm{j}}-2\sqrt{3}\hat{\mathrm{k}})$. The value of $x$ is

Unit vector in the direction of $\overrightarrow{P}\times \overrightarrow{Q} \text{is} \hat{n}=\frac{\overrightarrow{P}\times \overrightarrow{Q}}{|\overrightarrow{P}\times \overrightarrow{Q}|}$.

Here, $\overrightarrow{P}\times \overrightarrow{Q}=\left|\begin{array}{ccc}\hat{\mathrm{i}}& \hat{\mathrm{j}}& \hat{\mathrm{k}}\\ 3& \sqrt{3}& 2\\ 4& \sqrt{3}& 2.5\end{array}\right|=\sqrt{3}\frac{\hat{\mathrm{i}}}{2}+\frac{\hat{\mathrm{j}}}{2}-\sqrt{3}\hat{\mathrm{k}}$

And $\left|\overrightarrow{P}\times \overrightarrow{Q}\right|=\sqrt{{\left(\frac{\sqrt{3}}{2}\right)}^2+{\left(\frac{1}{2}\right)}^2+{\left(\sqrt{3}\right)}^2}=\sqrt{4}=2$

$\Rightarrow \frac{\overrightarrow{P}\times \overrightarrow{Q}}{|\overrightarrow{P}\times \overrightarrow{Q}|}=\frac{1}{2}\left(\sqrt{3}\frac{\hat{\mathrm{i}}}{2}+\frac{\hat{\mathrm{j}}}{2}-\sqrt{3}\hat{\mathrm{k}}\right)$

$$\begin{gathered} =\frac{1}{4}(\sqrt{3}\hat{\mathrm{i}}+\hat{\mathrm{j}}-2\sqrt{3}\hat{\mathrm{k}}) \\ \Rightarrow \mathrm{x}=4 \end{gathered}$$