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Three Dimensional Geometry

Question
CBSEENMA12033469

Find the angle between the line fraction numerator straight x minus 1 over denominator 1 end fraction space equals space fraction numerator straight y minus 2 over denominator negative 1 end fraction space equals space fraction numerator straight z plus 1 over denominator 1 end fraction and the plane 2x – y + z= 1. 

Solution
The equation of line is
fraction numerator straight x minus 1 over denominator 1 end fraction space space equals fraction numerator straight y minus 2 over denominator negative 1 end fraction space equals space fraction numerator straight z plus 1 over denominator 1 end fraction
Its direction ratios are 1, – 1, 1
The equation of plane is
2x –y + z = 1
∴ direction ratios of the normal to the plane are 2, – 1, 1.
Let θ be the angle between the line and plane.
therefore space space space space sin space straight theta space equals space fraction numerator left parenthesis 1 right parenthesis thin space left parenthesis 2 right parenthesis space plus space left parenthesis negative 1 right parenthesis thin space left parenthesis negative 1 right parenthesis thin space plus space left parenthesis 1 right parenthesis thin space left parenthesis 1 right parenthesis over denominator square root of 1 plus 1 plus 1 end root space square root of 4 plus 1 plus 1 end root end fraction space equals space fraction numerator 2 plus 1 plus 1 over denominator square root of 3 space square root of 6 end fraction space equals space fraction numerator 4 over denominator square root of 18 end fraction
                 equals space fraction numerator 4 over denominator square root of 9 cross times 2 end root end fraction space equals space fraction numerator 4 over denominator 3 square root of 2 end fraction space equals space fraction numerator 2 square root of 2 over denominator 3 end fraction
therefore space space space space space space space straight theta space equals space sin to the power of negative 1 end exponent open parentheses fraction numerator 2 space square root of 2 over denominator 3 end fraction close parentheses