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

Answer: 4

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