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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
$3$
$2$
$1$
$4$
Solution
$\begin{array}{l}\overrightarrow{ P } \times \overrightarrow{ Q }=\left|\begin{array}{ccc}\hat{ i } & \hat{ j } & \hat{ k } \\ 3 & \sqrt{3} & 2 \\ 4 & \sqrt{3} & 2.5\end{array}\right|=\sqrt{3} \frac{\hat{ i }}{2}+\frac{\hat{ j }}{2}-\sqrt{3} \hat{ k } \\ \Rightarrow \frac{\overrightarrow{ P } \times \overrightarrow{ Q }}{|\overrightarrow{ P } \times \overrightarrow{ Q }|}=\frac{1}{2}\left(\sqrt{3} \frac{\hat{ i }}{2}+\frac{\hat{ j }}{2}-\sqrt{3} \hat{ k }\right) \\ =\frac{1}{4}(\sqrt{3} \hat{ i }+\hat{ j }-2 \sqrt{3} \hat{ k }) \quad x =4\end{array}$
