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Three Dimensional Geometry

Question
CBSEENMA12033293

Find the angle between the following pair of lines:
fraction numerator 5 minus straight x over denominator 3 end fraction space equals space fraction numerator straight y plus 3 over denominator negative 2 end fraction comma space space straight z space equals space 5 space space space and space space straight x over 1 space equals space fraction numerator 1 minus straight y over denominator 3 end fraction space equals fraction numerator straight z minus 5 over denominator 2 end fraction

Solution
The equations of two lines are
                            fraction numerator 5 minus straight x over denominator 3 end fraction space equals space fraction numerator straight y plus 3 over denominator negative 2 end fraction comma space space space straight z space equals space 5 space space space and space space space straight x over 1 space equals space fraction numerator 1 minus straight y over denominator 3 end fraction space equals space fraction numerator straight z minus 5 over denominator 2 end fraction
or                    fraction numerator straight x minus 5 over denominator negative 3 end fraction space equals space fraction numerator straight y plus 3 over denominator negative 2 end fraction space equals space fraction numerator straight z minus 5 over denominator 0 end fraction space space space and space space space straight x over 1 space equals space fraction numerator straight y minus 1 over denominator negative 3 end fraction space equals space fraction numerator straight z minus 5 over denominator 2 end fraction
Direction ratios of two lines are –3, –2, 0 and 1, –3, 2.
Let θ be angle between the lines
therefore space space space space space cos space straight theta space equals space fraction numerator left parenthesis negative 3 right parenthesis thin space left parenthesis 1 right parenthesis space plus space left parenthesis negative 2 right parenthesis thin space left parenthesis negative 3 right parenthesis space plus space left parenthesis 0 right parenthesis thin space left parenthesis 2 right parenthesis over denominator square root of 9 plus 4 plus 0 end root space square root of 1 plus 9 plus 4 end root end fraction space equals space fraction numerator negative 3 plus 6 plus 0 over denominator square root of 13 space square root of 14 end fraction space
therefore space space space space space space cos space straight theta space equals space fraction numerator 3 over denominator square root of 182 end fraction space space space space space space space space rightwards double arrow space space space space space straight theta space equals space cos to the power of negative 1 end exponent open parentheses fraction numerator 3 over denominator square root of 182 end fraction close parentheses

Some More Questions From Three Dimensional Geometry Chapter

Find the direction cosines of x, y and z-axis.