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Question
The direction ratios of the line perpendicular to the lines \[\frac{x - 7}{2} = \frac{y + 17}{- 3} = \frac{z - 6}{1} \text{ and }, \frac{x + 5}{1} = \frac{y + 3}{2} = \frac{z - 4}{- 2}\] are proportional to
Options
(a) 4, 5, 7
(b) 4, −5, 7
(c) 4, −5, −7
(d) −4, 5, 7
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Solution
(a) 4, 5, 7
We have ,
\[\frac{x - 7}{2} = \frac{y + 17}{- 3} = \frac{z - 6}{1} \]
\[\frac{x + 5}{1} = \frac{y + 3}{2} = \frac{z - 4}{- 2}\]
The direction ratios of the given lines are proportional to 2,-3,1 and 1,2,-2.
The vectors parallel to the given vectors are \[\overrightarrow{b_1} = 2 \hat{i} - 3 \hat{j} + \hat{k} \text{ and } \overrightarrow{b_2} = \hat{i} + 2 \hat{j} - 2 \hat{k} \]
Vector perpendicular to the given two lines is ,
\[\overrightarrow{b} = \overrightarrow{b_1} \times \overrightarrow{b}_2 \]
\[ = \begin{vmatrix}\hat{i} & \hat{j} & \hat{k} \\ 2 & - 3 & 1 \\ 1 & 2 & - 2\end{vmatrix}\]
\[ = 4 \hat{i} + 5 \hat{j} + 7 \hat{k} \]
Hence, the direction ratios of the line perpendicular to the given two lines are proportional to 4, 5, 7.
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