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Three Dimensional Geometry

Question
CBSEENMA12033287

Find the angle between the pair of lines with direction ratios 2, 2, 1 and 4, 1, 8.

Solution
The direction ratios of the two lines are 2, 2, 1 and 4, 1, 8.
Let θ be the angle between the two lines
therefore space space space space space space space cos space straight theta space equals space fraction numerator left parenthesis 2 right parenthesis thin space left parenthesis 4 right parenthesis space plus space left parenthesis 2 right parenthesis thin space left parenthesis 1 right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis 8 right parenthesis over denominator square root of left parenthesis 2 right parenthesis squared plus left parenthesis 2 right parenthesis squared plus left parenthesis 1 right parenthesis end root space square root of left parenthesis 4 right parenthesis squared plus left parenthesis 1 right parenthesis squared plus left parenthesis 8 right parenthesis squared end root end fraction
                                     open square brackets because space space cos space straight theta space equals space fraction numerator straight a subscript 1 straight a subscript 2 plus straight b subscript 1 straight b subscript 2 plus straight c subscript 1 straight c subscript 2 over denominator square root of straight a subscript 1 squared plus straight b subscript 1 squared plus straight c subscript 1 squared end root space space square root of straight a subscript 2 squared plus straight b subscript 2 squared plus straight c subscript 2 squared end root end fraction close square brackets
                     equals space fraction numerator 8 plus 2 plus 8 over denominator 3 cross times 9 end fraction space equals fraction numerator 18 over denominator 3 cross times 9 end fraction space equals 2 over 3
therefore                 straight theta space equals space cos to the power of negative 1 end exponent space open parentheses 2 over 3 close parentheses