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Question
Find the equation of the line passing through the point (1, −1, 1) and perpendicular to the lines joining the points (4, 3, 2), (1, −1, 0) and (1, 2, −1), (2, 1, 1).
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Solution
The direction ratios of the line joining the points (4, 3, 2), (1, -1, 0) and (1, 2, -1), (2, 1, 1) are -3,-4,-2 and 1, -1,2 ,respectively.
Let:
\[\overrightarrow{b_1} = - 3 \hat{i} - 4 \hat{j} - 2 \hat{k} \]
\[ \overrightarrow{b_2} = \hat{i} - \hat{j} + 2 \hat{k}\]
Since the required line is perpendicular to the lines parallel to the vectors
\[\overrightarrow{b_1} = - 3 \hat{i} - 4 \hat{j} - 2 \hat{k} \text{ and } \overrightarrow{b_2} = \hat{i} - \hat{j} + 2 \hat{k} \] it is parallel to the vector
\[\overrightarrow{b} = \overrightarrow{b_1} \times \overrightarrow{b_2}\]
Now,
\[\overrightarrow{b} = \overrightarrow{b_1} \times \overrightarrow{b_2} \]
\[ = \begin{vmatrix}\hat{i} & \hat{j} & \hat{k} \\ - 3 & - 4 & - 2 \\ 1 & - 1 & 2\end{vmatrix}\]
\[ = - 10 \hat{i} + 4 \hat{j} + 7 \hat{k}\]
So, the direction ratios of the required line are proportional to
-10, 4, 7.
The equation of the required line passing through the point (1,-1, 1) and having direction ratios proportional to -10, 4, 7 is
\[\frac{x - 1}{- 10} = \frac{y + 1}{4} = \frac{z - 1}{7}\]
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