यदि \(\overrightarrow{\mathrm{P}}=3 \hat{\mathrm{i}}+\sqrt{3} \hat{\mathrm{j}}+2 \hat{\mathrm{k}}\) एवं \(\overrightarrow{\mathrm{Q}}=4 \hat{\mathrm{i}}+\sqrt{3} \hat{\mathrm{j}}+2.5 \hat{\mathrm{k}}\) तो \(\overrightarrow{\mathrm{P}} \times \overrightarrow{\mathrm{Q}}\) की दिशा में इकाई सदिश \(\frac{1}{\mathrm{x}}(\sqrt{3 \hat{\mathrm{i}}}+\hat{\mathrm{j}}-2 \sqrt{3} \hat{\mathrm{k}})\) है, तो \(\mathrm{x}\) का मान है.............।

If \(\overrightarrow{ P }=3 \hat{ i }+\sqrt{3} \hat{ j }+2 \hat{ k }\) and \(\overrightarrow{ Q }=4 \hat{ i }+\sqrt{3} \hat{ j }+2.5 \hat{ k }\) then, The unit vector in the direction of \(\overrightarrow{ P } \times \overrightarrow{ Q }\) is \(\frac{1}{x}(\sqrt{3} \hat{i}+\hat{j}-2 \sqrt{3} \hat{k})\). The value of \(x\) is
\(\begin{array}{l}\overrightarrow{ P } \times \overrightarrow{ Q }=\left|\begin{array}{ccc}\hat{ i } & \hat{ j } & \hat{ k } \\ 3 & \sqrt{3} & 2 \\ 4 & \sqrt{3} & 2.5\end{array}\right|=\sqrt{3} \frac{\hat{ i }}{2}+\frac{\hat{ j }}{2}-\sqrt{3} \hat{ k } \\ \Rightarrow \frac{\overrightarrow{ P } \times \overrightarrow{ Q }}{|\overrightarrow{ P } \times \overrightarrow{ Q }|}=\frac{1}{2}\left(\sqrt{3} \frac{\hat{ i }}{2}+\frac{\hat{ j }}{2}-\sqrt{3} \hat{ k }\right)\end{array}\)
\( =\frac{1}{4}(\sqrt{3} \hat{ i }+\hat{ j }-2 \sqrt{3} \hat{ k })\)
The unit vector in the direction of \(\overrightarrow{ P } \times \overrightarrow{ Q }\) is \(\frac{1}{x}(\sqrt{3} \hat{i}+\hat{j}-2 \sqrt{3} \hat{k})\)
hence \( x=4\)