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Three Dimensional Geometry

Question
CBSEENMA12033470

Find the angle between the line fraction numerator straight x plus 1 over denominator 3 end fraction space equals space fraction numerator straight y minus 1 over denominator 2 end fraction space equals space fraction numerator straight z minus 2 over denominator 4 end fraction and the plane 2x + y – 3z + 4 = 0.

Solution
The equation of line is fraction numerator straight x plus 1 over denominator 3 end fraction space equals space fraction numerator straight y minus 1 over denominator 2 end fraction space equals space fraction numerator straight z minus 2 over denominator 4 end fraction
Its direction ratios are 3, 2, 4.
The equation of plane is 2x + y – 3z + 4 = 0
∴ direction ratios of the normal to the plane are 2, 1, –3.
Let θ be the angle between the line and plane
          therefore space space space space space sin space straight theta space equals space fraction numerator left parenthesis 3 right parenthesis thin space left parenthesis 2 right parenthesis space plus space left parenthesis 2 right parenthesis thin space left parenthesis 1 right parenthesis space plus space left parenthesis 4 right parenthesis thin space left parenthesis negative 3 right parenthesis over denominator square root of 9 plus 4 plus 16 end root space square root of 4 plus 1 plus 9 end root end fraction space equals space fraction numerator 6 plus 2 minus 12 over denominator square root of 29 space square root of 14 end fraction space equals space fraction numerator negative 4 over denominator square root of 29 space square root of 14 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 negative fraction numerator 4 over denominator square root of 29 space square root of 14 end fraction close parentheses.

Some More Questions From Three Dimensional Geometry Chapter

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