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Three Dimensional Geometry

Question
CBSEENMA12033472

Find the angle between the line fraction numerator straight x minus 2 over denominator 3 end fraction space equals space fraction numerator straight y plus 1 over denominator negative 1 end fraction space equals space fraction numerator straight z minus 3 over denominator 2 end fraction and the plane 3x + 4y + z + 5 = 0.

Solution
The equation of line is
fraction numerator straight x minus 2 over denominator 3 end fraction space equals fraction numerator straight y plus 1 over denominator negative 1 end fraction space equals space fraction numerator straight z minus 3 over denominator 2 end fraction
Its direction ratios are 3, – 1, 2.
The equation of plane is 3.x + 4y + z + 5 = 0
∴  direction ratios of the normal to the plane are 3, 4, 1.
Let θ be the angle between the line and the plane
therefore space space sin space straight theta space equals space fraction numerator left parenthesis 3 right parenthesis thin space left parenthesis 3 right parenthesis space plus space left parenthesis negative 1 right parenthesis thin space left parenthesis 4 right parenthesis space plus space left parenthesis 2 right parenthesis thin space left parenthesis thin space 1 right parenthesis over denominator square root of 9 plus 1 plus 4 end root space square root of 9 plus 16 plus 1 end root end fraction space equals space fraction numerator 7 over denominator square root of 14 space square root of 26 end fraction space equals space fraction numerator 7 over denominator square root of 2 space square root of 7 space. space square root of 2 space square root of 13 end fraction
               equals space fraction numerator square root of 7 over denominator 2 square root of 13 end fraction space equals space 1 half square root of 7 over 13 end root
rightwards double arrow space space space space straight theta space equals space sin to the power of negative 1 end exponent open parentheses 1 half square root of 7 over 13 end root close parentheses

Some More Questions From Three Dimensional Geometry Chapter

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