Let a line L pass through the point P(2, 3, 1) and be parallel to the line x + 3 y - 2 z - 2 = 0 = x - y + 2 z . If the distance of L from the point (5, 3, 8) is , then 3 2 is equal to _________ . [2023]

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Solution

Q.
Let a line L pass through the point P(2, 3, 1) and be parallel to the line $x + 3 y - 2 z - 2 = 0 = x - y + 2 z .$ If the distance of L from the point (5, 3, 8) is $α$ , then $3 α^{2}$ is equal to _________ . [2023]

Ans.
(158)

$Let \vec{a} = \hat{i} + 3 \hat{j} - 2 \hat{k} and \vec{b} = \hat{i} - \hat{j} + 2 \hat{k}$

$\vec{a} \times \vec{b} =$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 3 & - 2 \\ 1 & - 1 & 2 \end{matrix} |$ $= 4 \hat{i} - 4 \hat{j} - 4 \hat{k} = 4 (\hat{i} - \hat{j} - \hat{k})$

$Line will be parallel to \vec{a} \times \vec{b} \Rightarrow \vec{n} = \hat{i} - \hat{j} - \hat{k}$

$Equation of the line passing through the point P (2, 3, 1) and parallel to \vec{a} \times \vec{b}$

$∴$ $\frac{x - 2}{1} = \frac{y - 3}{- 1} = \frac{z - 1}{- 1}$

${\vec{a}}_{1} = 2 \hat{i} + 3 \hat{j} + \hat{k}, {\vec{a}}_{2} = 5 \hat{i} + 3 \hat{j} + 8 \hat{k}$ $\Rightarrow {\vec{a}}_{2} - {\vec{a}}_{1} = 3 \hat{i} + 7 \hat{k}$

$({\vec{a}}_{2} - {\vec{a}}_{1}) \times \vec{n} =$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 0 & 7 \\ 1 & - 1 & - 1 \end{matrix} |$ $= 7 \hat{i} - \hat{j} (- 3 - 7) + \hat{k} (- 3)$

$= 7 \hat{i} + 10 \hat{j} - 3 \hat{k}$

$Now, α =$ $| \frac{({\vec{a}}_{2} - {\vec{a}}_{1}) \times \vec{n}}{| \vec{n} |} |$ $= \frac{| 7 \hat{i} + 10 \hat{j} - 3 \hat{k} |}{\sqrt{1 + 1 + 1}}$

$= \frac{\sqrt{49 + 100 + 9}}{\sqrt{3}} = \frac{\sqrt{158}}{\sqrt{3}} \Rightarrow 3 α^{2} = 158 units$

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