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Three Dimensional Geometry

Question
CBSEENMA12033406
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Find the angle between the planes whose vector equations are
straight r with rightwards arrow on top. space open parentheses straight i with hat on top space plus space straight j with hat on top space minus space 2 space straight k with hat on top close parentheses space equals space 3 space space and space space space straight r with rightwards arrow on top. space space open parentheses 2 straight i with hat on top space minus space 2 straight j with hat on top space plus space straight k with hat on top close parentheses space equals space 2

Solution
The equation of the two planes are
straight r with rightwards arrow on top. space open parentheses straight i with hat on top space plus space straight j with hat on top space minus space 2 space straight k with hat on top close parentheses space equals space 3 space space space and space straight r with rightwards arrow on top. space open parentheses 2 space straight i with hat on top space minus space 2 space straight j with hat on top space plus space straight k with hat on top close parentheses space equals space 2
therefore space space space space space space space space stack straight n subscript 1 with rightwards arrow on top space equals space straight i with hat on top space plus space straight j with hat on top space minus space 2 space straight k with hat on top comma space space space stack straight n subscript 2 with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space 2 space straight j with hat on top space plus space straight k with hat on top therefore space space space space space space open vertical bar stack straight n subscript 1 with rightwards arrow on top close vertical bar space equals space square root of 1 plus 1 plus 4 end root space equals square root of 6 comma space space open vertical bar stack straight n subscript 2 with rightwards arrow on top close vertical bar space equals square root of 4 plus 4 plus 1 end root space space equals square root of 9 space equals space 3 space space space space space space space space stack straight n subscript 1 with rightwards arrow on top. space stack straight n subscript 2 with rightwards arrow on top space equals space left parenthesis 1 right parenthesis thin space left parenthesis 2 right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis negative 2 right parenthesis space plus space left parenthesis negative 2 right parenthesis thin space left parenthesis 1 right parenthesis space equals space 2 minus 2 minus 2 space equals space minus 2
Let θ be the angle between the planes.
therefore space space space cos space straight theta space equals space fraction numerator stack straight n subscript 1 with rightwards arrow on top. space stack straight n subscript 2 with rightwards arrow on top over denominator open vertical bar stack straight n subscript 1 with rightwards arrow on top close vertical bar space open vertical bar stack straight n subscript 2 with rightwards arrow on top close vertical bar end fraction space equals fraction numerator negative 2 over denominator 3 square root of 6 end fraction therefore space space space space space acute space angle space straight theta space is space given space by space space space space space space space space space space space space space cos space straight theta space equals space fraction numerator 2 over denominator 3 square root of 6 end fraction therefore space space space space space space space straight theta space equals cos to the power of negative 1 end exponent open parentheses fraction numerator 2 over denominator 3 square root of 6 end fraction close parentheses

Some More Questions From Three Dimensional Geometry Chapter

If α, β, γ are the angles which a line makes with the axes, prove that sin2α + sin2 β + sin2γ = 2.

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