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Find the angle between the pair of lines whose direction ratios are (3, 5, 4) and (1, 1, 2) respectively.
1. \({\sin ^{ - 1}}\left( {\frac{{8\sqrt 3 }}{{15}}} \right)\)
2. \({\cos ^{ - 1}}\left( {\frac{{8\sqrt 3 }}{{15}}} \right)\)
3. \({\cos ^{ - 1}}\left( {\frac{{16\sqrt 3 }}{{15}}} \right)\)
4. \({\sin ^{ - 1}}\left( {\frac{{16\sqrt 3 }}{{15}}} \right)\)
5. None of these

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Correct Answer - Option 2 : \({\cos ^{ - 1}}\left( {\frac{{8\sqrt 3 }}{{15}}} \right)\)

Let L1 and L2 be two lines passing through the origin and with direction ratios a1, b1, c1 and a2, b2, c2, respectively. The angle θ between them is given by

\(\begin{array}{l} \cos \theta = \left| {\frac{{{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2}}}{{\sqrt {a_1^2 + b_1^2 + c_1^2} \sqrt {a_2^2 + b_2^2 + c_2^2} }}} \right|\\ \cos \theta = \left| {\frac{{3.1 + 5.1 + 4.2}}{{\sqrt {{3^2} + {5^2} + {4^2}} \sqrt {{1^2} + {1^2} + {2^2}} }}} \right| = \frac{{16}}{{\sqrt {50} \sqrt 6 }} = \frac{{16}}{{5\sqrt 2 \sqrt 6 }} = \frac{{16}}{{5\sqrt {12} }}\\ \cos \theta = \frac{{16 \times \sqrt {12} }}{{5 \times 12}} = \frac{{16 \times 2\sqrt 3 }}{{5 \times 12}} = \frac{{8\sqrt 3 }}{{15}} \end{array}\)

Hence, the required angle is \(\;{\cos ^{ - 1}}\left( {\frac{{8\sqrt 3 }}{{15}}} \right)\).

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