Question

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

Answer 1

$ \vec{ P } \times \vec{ Q }= \begin{vmatrix}\hat{ i } & \hat{ j } & \hat{ k } \\ 3 & \sqrt{3} & 2 \\ 4 & \sqrt{3} & 2.5\end{vmatrix}=\sqrt{3} \frac{\hat{ i }}{2}+\frac{\hat{ j }}{2}-\sqrt{3} \hat{ k } $
$\Rightarrow \frac{\vec{ P } \times \vec{ Q }}{|\vec{ P } \times \vec{ 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 }) x =4$