Q 11-03-090JEE MainJEE Main 2023 (25 Jan, Shift 1)Medium
If $\vec P=3\hat i+\sqrt3\hat j+2\hat k$ and $\vec Q=4\hat i+\sqrt3\hat j+2.5\hat k$, then the unit vector in the direction of $\vec P\times\vec Q$ is $\dfrac1x\left(\sqrt3\hat i+\hat j-2\sqrt3\hat k\right)$. The value of $x$ is ______.
Numerical value type. Enter your answer.
Answer: 4
$$\vec P\times\vec Q=\hat i(2.5\sqrt3-2\sqrt3)-\hat j(7.5-8)+\hat k(3\sqrt3-4\sqrt3)=\frac{\sqrt3}{2}\hat i+\frac12\hat j-\sqrt3\hat k$$
Magnitude $=\sqrt{\tfrac34+\tfrac14+3}=2$, so the unit vector is $\dfrac14\left(\sqrt3\hat i+\hat j-2\sqrt3\hat k\right)$ and $x=4$.
Solution by Sreeraj P, M.Sc Physics