If P → = 3 i ^ + 3 j ^ + 2 k ^ and Q → = 4 i ^ + 3 j ^ + 2 . 5 k ^ then, the unit vector in the…
If and then, the unit vector in the direction of is . The value of is
Solution
Unit vector in the direction of $\vec{P} \times \vec{Q}$ is $\vec{n} = \frac{\vec{P} \times \vec{Q}}{|\vec{P} \times \vec{Q}|}$.
Here, $\vec{P} \times \vec{Q} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & \sqrt{3} & 2 \\ 4 & \sqrt{3} & 2.5 \end{vmatrix} = \sqrt{3} \hat{i} + \hat{j} - \sqrt{3} \hat{k}$.
And $|\vec{P} \times \vec{Q}| = \sqrt{\left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{1}{2}\right)^2 + (\sqrt{3})^2} = \sqrt{4} = 2$.
$\Rightarrow \frac{\vec{P} \times \vec{Q}}{|\vec{P} \times \vec{Q}|} = \frac{1}{2} \left(\sqrt{3} \hat{i} + \hat{j} - \sqrt{3} \hat{k}\right)$.
$= \frac{1}{4} (\sqrt{3} \hat{i} + \hat{j} - 2\sqrt{3} \hat{k}) \Rightarrow x = 4$.
Asked in: JEE Main 2023 (25 Jan Shift 1)
Practice more Mathematical Methods questions on Aicharya