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Three Dimensional Geometry

Question
CBSEENMA12033471

Find the angle between the lines fraction numerator straight x plus 1 over denominator 2 end fraction space equals space straight y over 3 space equals space fraction numerator straight z minus 3 over denominator 6 end fraction and the plane 10x + 2y – 11z = 3.

Solution
The equation of line is
         fraction numerator straight x plus 1 over denominator 2 end fraction space equals space straight y over 3 space equals space fraction numerator straight z minus 3 over denominator 6 end fraction
Its direction ratios are 2, 3, 6.
The equation of plane is
10x + 2y – 11 z = 3
∴  direction ratios of the normal to the plane are 10, 2, –11.
Let θ be the angle between the line and plane.
therefore space space space sin space straight theta space equals space fraction numerator open vertical bar left parenthesis 2 right parenthesis thin space left parenthesis 10 right parenthesis space plus space left parenthesis 3 right parenthesis thin space left parenthesis 2 right parenthesis space plus space left parenthesis 6 right parenthesis thin space left parenthesis negative 11 right parenthesis close vertical bar over denominator square root of 4 plus 9 plus 36 end root space square root of 100 plus 4 plus 121 end root end fraction
space space space space space space space space space space space space space space space space space space equals space fraction numerator open vertical bar 20 plus 6 minus 66 close vertical bar over denominator square root of 49 space square root of 225 end fraction space equals space fraction numerator open vertical bar negative 40 close vertical bar over denominator 7 cross times 15 end fraction space equals fraction numerator 40 over denominator 7 cross times 15 end fraction
therefore space space space space space sin space straight theta space equals space 8 over 21 space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space straight theta space equals space sin to the power of negative 1 end exponent open parentheses 8 over 21 close parentheses.

Some More Questions From Three Dimensional Geometry Chapter

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