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Question
Find the angle between two lines, one of which has direction ratios 2, 2, 1 while the other one is obtained by joining the points (3, 1, 4) and (7, 2, 12).
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Solution
The direction ratios of the line joining the points (3, 1, 4) and (7, 2, 12) are proportional to 4, 1, 8.
Let
\[\overrightarrow{m_1} \text{ and } \overrightarrow{m_2}\] be vectors parallel to the lines having direction ratios proportional to 2, 2, 1 and 4, 1, 8.
Now,
\[\overrightarrow{b_1} = 2 \hat{i} + 2 \hat{j} + \hat{k} \]
\[ \overrightarrow{b_2} = 4 \hat{i} + \hat{j} + 8 \hat{k} \]
If θ is the angle between the given lines, then
\[\cos \theta = \frac{\overrightarrow{m_1} . \overrightarrow{m_2}}{\left| \overrightarrow{m_1} \right| \left| \overrightarrow{m_2} \right|}\]
\[ = \frac{\left( 2 \hat{i} + 2 \hat{j} + \hat{k} \right) . \left( 4 \hat{i} + \hat{j} + 8 \hat{k} \right)}{\sqrt{2^2 + 2^2 + 1^2} \sqrt{4^2 + 1^2 + 8^2}}\]
\[ = \frac{8 + 2 + 8}{3 \times 9}\]
\[ = \frac{2}{3}\]
\[ \Rightarrow \theta = \cos^{- 1} \left( \frac{2}{3} \right)\]
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