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Three Dimensional Geometry

Question
CBSEENMA12033403

Find the angle between the two planes
3 x – 6 y + 2 z = 7 and 2 x + 2 y – 2 z = 5.

Solution

The equations of given planes are
3 x – 6 + 2 z = 7
and 2 x + 2 y – 2 z = 5
∴  a= 3, b1 = – 6, c= 2
and a2 = 2, b2 = 2, c2 = – 2
Now,      
             cos space straight theta space equals space open vertical bar fraction numerator straight a subscript 1 straight a subscript 2 plus straight b subscript 1 straight b subscript 2 plus straight c subscript 1 straight c subscript 2 over denominator square root of straight a subscript 1 squared plus straight b subscript 1 squared plus straight c subscript 1 squared end root space square root of straight a subscript 2 squared plus straight b subscript 2 squared plus straight c subscript 2 squared end root end fraction close vertical bar
space space space space space space space space space space equals open vertical bar fraction numerator left parenthesis 3 right parenthesis thin space left parenthesis 2 right parenthesis space plus left parenthesis negative 6 right parenthesis thin space left parenthesis 2 right parenthesis space plus space left parenthesis 2 right parenthesis thin space left parenthesis negative 2 right parenthesis over denominator square root of left parenthesis 3 right parenthesis squared plus left parenthesis negative 6 right parenthesis squared plus left parenthesis 2 right parenthesis squared end root space square root of left parenthesis 2 right parenthesis squared plus left parenthesis 2 right parenthesis squared plus left parenthesis negative 2 right parenthesis squared end root end fraction close vertical bar
space space space space space space space space space space equals open vertical bar fraction numerator 6 minus 12 minus 4 over denominator square root of 9 plus 36 plus 4 end root space square root of 4 plus 4 plus 4 end root end fraction close vertical bar space equals space open vertical bar fraction numerator negative 10 over denominator square root of 49 space square root of 12 end fraction close vertical bar
space space space space space space space space space space space equals space fraction numerator 10 over denominator 7 cross times 2 square root of 3 end fraction space equals fraction numerator 5 over denominator 7 square root of 3 end fraction space equals fraction numerator 5 square root of 3 over denominator 21 end fraction
therefore space space space space straight theta space equals space cos to the power of negative 1 end exponent open parentheses fraction numerator 5 square root of 3 over denominator 21 end fraction close parentheses.

Some More Questions From Three Dimensional Geometry Chapter

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