The distance of the point P(4, 6, ?2) from the line passing through the point (?3, 2, 3) and parallel to a line with direction ratios 3, 3, ?1 is equal to: [2023]

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Solution

Q.
The distance of the point P(4, 6, −2) from the line passing through the point (−3, 2, 3) and parallel to a line with direction ratios 3, 3, −1 is equal to: [2023]

1 $2 \sqrt{3}$

2 $\sqrt{14}$

3 $3$

4 $\sqrt{6}$

Ans.
(2)

$Let Q = - 3 \hat{i} + 2 \hat{j} + 3 \hat{k}$

$\vec{P Q} = - 7 \hat{i} - 4 \hat{j} + 5 \hat{k}, \vec{b} = 3 \hat{i} + 3 \hat{j} - \hat{k} \Rightarrow$ $| \vec{b} |$ $= \sqrt{19}$

$\vec{P Q} \times \vec{b} =$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ - 7 & - 4 & 5 \\ 3 & 3 & - 1 \end{matrix} |$ $= - 11 \hat{i} + 8 \hat{j} - 9 \hat{k}$

$Required distance =$ $| \frac{\vec{P Q} \times \vec{b}}{| \vec{b} |} |$ $= \frac{\sqrt{11^{2} + 8^{2} + 9^{2}}}{\sqrt{19}} = \sqrt{14} units$

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