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Question
Find the angle between the following pair of line:
\[\frac{x + 4}{3} = \frac{y - 1}{5} = \frac{z + 3}{4} \text { and } \frac{x + 1}{1} = \frac{y - 4}{1} = \frac{z - 5}{2}\]
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Solution
\[\frac{x + 4}{3} = \frac{y - 1}{5} = \frac{z + 3}{4} and \frac{x + 1}{1} = \frac{y - 4}{1} = \frac{z - 5}{2}\]
Let
\[\overrightarrow{b_1}\] and \[\overrightarrow{b_2}\] be vectors parallel to the given line.
\[\overrightarrow{b_1}\] = 3 \hat{i} + 5 \hat{j} + 4 \hat{k} \]
\[\overrightarrow{b_2}\] = \hat{i} + \hat{j}+ 2 \hat{k} \]
If θ is the angle between the given line, then
\[\cos \theta = \frac{\overrightarrow{b_1} . \overrightarrow{b_2}}{\left| \overrightarrow{b_1} \right| \left| \overrightarrow{b_2} \right|}\]
\[ = \frac{\left( 3 \hat{i} + 5 \hat{j} + 4 \hat{k} \right) . \left( \hat{i} + \hat{j} + 2 \hat{k} \right)}{\sqrt{3^2 + 5^2 + 4^2} \sqrt{1^2 + 1^2 + 2^2}}\]
\[ = \frac{3 + 5 + 8}{10\sqrt{3}}\]
\[ = \frac{8}{5\sqrt{3}}\]
\[ \Rightarrow \theta = \cos^{- 1} \left( \frac{8}{5\sqrt{3}} \right)\]
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