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Question
Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, −1) and (4, 3, −1).
Solution
The direction ratios of the line joining the origin to the point (2, 1, 1) are 2, 1, 1.
Let
\[\overrightarrow{b_1} = 2 \hat{i} + \hat{j} + \hat{k}\]
The direction ratios of the line joining the points (3, 5,-1) and (4,3,-1) are 1, -2,0
Let
\[\overrightarrow{b_2} = \hat{i} - 2 \hat{j} + 0 \hat{k}\]
Now,
\[\overrightarrow{b_1} . \overrightarrow{b_2} = \left( 2 \hat{i} + \hat{j} + \hat{k } \right) . \left( \hat{i} - 2 \hat{j} + 0 \hat{k} \right)\]
\[ = 2 - 2 + 0\]
\[ = 0\]
\[ \therefore \overrightarrow{b_1} \perp \overrightarrow{b_2}\]
Hence, the two lines joining the given points are perpendicular to each other.
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