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Question
Show that the lines \[\frac{x - 5}{7} = \frac{y + 2}{- 5} = \frac{z}{1} \text{ and } \frac{x}{1} = \frac{y}{2} = \frac{z}{3}\] are perpendicular to each other.
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Solution
The direction ratios of the lines
\[\frac{x - 5}{7} = \frac{y + 2}{- 5} = \frac{z}{1} and \frac{x}{1} = \frac{y}{2} = \frac{z}{3}\] are proportional to 7, -5, 1 and 1, 2, 3, respectively.
Let:
\[\overrightarrow{b_1} = 7 \hat{i} - 5 \hat{j} + \hat{k} \]
\[ \overrightarrow{b_2} = \hat{i} + 2 \hat{j} + 3 \hat{k} \]
Now,
\[\overrightarrow{b_1} . \overrightarrow{b_2} = \left( 7 \hat{i} - 5 \hat{j} + \hat{k} \right) . \left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right)\]
\[ = 7 - 10 + 3\]
\[ = 0\]
\[ \therefore \overrightarrow{b_1} \perp \overrightarrow{b_2}\]
Hence, the given lines are perpendicular to each other.
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