If and then, the unit vector in the direction of is . The value of 'x' is ________ . [2023]

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Solution

Q.
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 $\frac{1}{x} (\sqrt{3} \hat{i} + \hat{j} - 2 \sqrt{3} \hat{k})$ . The value of 'x' is ________ . [2023]

Ans.
(4)

$\bar{P} = 3 \hat{i} + \sqrt{3} \hat{j} + 2 \hat{k}$ , $\bar{Q} = 4 \hat{i} + \sqrt{3} \hat{j} + 2.5 \hat{k}$

$\bar{P} \times \bar{Q} =$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & \sqrt{3} & 2 \\ 4 & \sqrt{3} & 2.5 \end{matrix} |$

$= \hat{i} (2.5 \sqrt{3} - 2 \sqrt{3}) - \hat{j} (3 \times 2.5 - 8) + \hat{k} (3 \sqrt{3} - 4 \sqrt{3})$

$= 0.5 \sqrt{3} \hat{i} + 0.5 \hat{j} - \sqrt{3} \hat{k} = \frac{1}{2} (\sqrt{3} \hat{i} + \hat{j} - 2 \sqrt{3} \hat{k})$

$| \bar{P} \times \bar{Q} |$ $= \frac{1}{4} (\sqrt{3} \hat{i} + \hat{j} - 2 \sqrt{3} \hat{k})$

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