Mathematics In Physics Question 3
Question 3 - 25 January - Shift 1
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 $\overrightarrow{{}P} \times \overrightarrow{{}Q}$ is $\frac{1}{x}(\sqrt{3} \hat{i}+\hat{j}-2 \sqrt{3} \hat{k})$. The value of $x$ is
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Answer: (4)
Solution:
$\overrightarrow{{}P} \times \overrightarrow{{}Q}= \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & \sqrt{3} & 2 \\ 4 & \sqrt{3} & 2.5\end{vmatrix} =\sqrt{3} \frac{\hat{i}}{2}+\frac{\hat{j}}{2}-\sqrt{3} \hat{k}$
$\Rightarrow \frac{\overrightarrow{{}P} \times \overrightarrow{{}Q}}{|\overrightarrow{{}P} \times \overrightarrow{{}Q}|}=\frac{1}{2}(\sqrt{3} \frac{\hat{i}}{2}+\frac{\hat{j}}{2}-\sqrt{3} \hat{k})$
$=\frac{1}{4}(\sqrt{3} \hat{i}+\hat{j}-2 \sqrt{3} \hat{k}) \quad x=4$
