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Three Dimensional Geometry

Question
CBSEENMA12033298

Show that the lines
fraction numerator straight x minus 5 over denominator 7 end fraction space equals space fraction numerator straight y plus 2 over denominator negative 5 end fraction space equals straight z over 1 space and space straight x over 1 space equals space straight y over 2 space equals space straight z over 3 are prependicular to each other. 

Solution
The equations of two lines are
     fraction numerator straight x minus 5 over denominator 7 end fraction space equals space fraction numerator straight y plus 2 over denominator negative 5 end fraction space equals space straight z over 1 space and space straight x over 1 space equals straight y over 2 space equals space straight z over 3
∴   direction ratios of two lines are 7, –5, 1 and 1, 2, 3.
Let θ be the angle between the lines
therefore space space space space space cos space straight theta space equals space fraction numerator left parenthesis 7 right parenthesis thin space left parenthesis 1 right parenthesis space plus space left parenthesis negative 5 right parenthesis thin space left parenthesis 2 right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis 3 right parenthesis over denominator square root of 49 plus 25 plus 1 end root space square root of 1 plus 4 plus 9 end root end fraction space equals fraction numerator 7 minus 10 plus 3 over denominator square root of 75 space square root of 14 end fraction space equals space fraction numerator 0 over denominator square root of 75 space square root of 14 end fraction space equals space 0
therefore space space space space space space space space straight theta space equals space 90 degree
∴ two lines are perpendicular to each other.