Find the angle between the pair of lines

\(\frac{{x - 5}}{3} = \frac{{y + 2}}{5} = \frac{{z + 2}}{4}\)

And \(\frac{{x - 1}}{1} = \frac{{y - 3}}{1} = \frac{{z - 3}}{2}\)

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AAI ATC Junior Executive 25 March 2021 Official Paper (Shift 1)
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  1. \({\cos ^{ - 1}}\frac{{15}}{{8\sqrt 3 }}\)
  2. \({\cos ^{ - 1}}\frac{{2\sqrt 3 }}{{15}}\)
  3. \({\cos ^{ - 1}}\frac{{8\sqrt 3 }}{{15}}\)
  4. \({\cos ^{ - 1}}\frac{{15}}{{2\sqrt 3 }}\)

Answer (Detailed Solution Below)

Option 3 : \({\cos ^{ - 1}}\frac{{8\sqrt 3 }}{{15}}\)
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Detailed Solution

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CONCEPT:

The angle between the lines with direction ratios \(\left\langle {{a_1},\;{b_1},\;{c_1}} \right\rangle \) and \(\left\langle {{a_2},\;{b_2},\;{c_2}} \right\rangle \) is given by: \(\cos {\bf{\theta }} = \;\frac{{{{\bf{a}}_1}{{\bf{a}}_2} + \;{{\bf{b}}_1}{{\bf{b}}_2} + {{\bf{c}}_1}{{\bf{c}}_2}\;}}{{\sqrt {{\bf{a}}_{1\;}^2 + \;{\bf{b}}_1^2 + \;{\bf{c}}_1^2} \sqrt {{\bf{a}}_{2\;}^2 + \;{\bf{b}}_2^2 + \;{\bf{c}}_2^2} }}\)

CALCULATION:
Given: The direction ratios of two lines are (3,5, 4) and (1, 1, 2)
 
Here, a1 = 3, b1 = 5, c1 = 4, a2 = 1, b2 = 1 and c2 = 2
⇒ \(\cos \theta = \frac{{3 \times 1 +5 \times 1 + 4 \times 2}}{{\sqrt {{3^2} + {5^2} + {{\left( { 4} \right)}^2}} \times \sqrt {{1^2} + {{\left( { 1} \right)}^2} + {2^2}} }} = \frac{{8 }}{{5\sqrt 3}}\)
⇒ \(\theta = {\cos ^{ - 1}}\left( { \frac{{8 }}{{5\sqrt 3}}} \right) \)
\(⇒\theta = {\cos ^{ - 1}}\frac{{8\sqrt 3 }}{{15}}\)
Hence, option 3 is the correct answer.
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