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Question
Write the angle between the lines \[\frac{x - 5}{7} = \frac{y + 2}{- 5} = \frac{z - 2}{1} \text{ and } \frac{x - 1}{1} = \frac{y}{2} = \frac{z - 1}{3} .\]
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Solution
We have ,
\[\frac{x - 5}{7} = \frac{y + 2}{- 5} = \frac{z - 2}{1} \]
\[\frac{x - 1}{1} = \frac{y}{2} = \frac{z - 1}{3}\]
The given lines are parallel to the vectors
\[\overrightarrow{b_1} = 7 \hat{i} - 5 \hat{j} + \hat{k} \text{ and } \overrightarrow{b_2} = \hat{i} + 2 \hat{j} + 3 \hat{k} \]
Let ,
\[\theta\] be the angle between the given lines.
Now ,
\[\cos \theta = \frac{\overrightarrow{b_1} . \overrightarrow{b_2}}{\left| \overrightarrow{b_1} \right| \left| \overrightarrow{b_2} \right|}\]
\[ = \frac{\left( 7 \hat{i} - 5 \hat{j} + \hat{k} \right) . \left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right)}{\sqrt{7^2 + \left( - 5 \right)^2 + 1^2} \sqrt{1^2 + 2^2 + 3^2}}\]
\[ = \frac{7 - 10 + 3}{\sqrt{49 + 25 + 1} \sqrt{1 + 4 + 9}}\]
\[ = 0\]
\[ \Rightarrow \theta = \frac{\pi}{2}\]
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