Find the angle between the line $straight x over 4 space equals space fraction numerator straight y minus 2 over denominator 3 end fraction space equals space fraction numerator straight z minus 1 over denominator 6 end fraction$ and the plane x + 2y + 2 + 3 = 0

Solution

The equation of line is

straight x over 4 space equals space fraction numerator straight y minus 2 over denominator 3 end fraction space equals space fraction numerator straight z minus 1 over denominator 6 end fraction

Its direction ratios are 4, 3, 6
The equation of plane is
x + 2y + z + 3 = 0
∴ direction ratios of the normal to the plane are 1,2, 1.
Let θ be the angle between the line and plane

therefore space space space space sin space straight theta space equals space fraction numerator left parenthesis 4 right parenthesis thin space left parenthesis 1 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 1 right parenthesis over denominator square root of 16 plus 9 plus 36 end root space square root of 1 plus 4 plus 1 end root end fraction space equals space fraction numerator 4 plus 6 plus 6 over denominator square root of 61 space square root of 6 end fraction space equals space fraction numerator 16 over denominator square root of 366 end fraction therefore space space space space space space space space straight theta space equals space sin to the power of negative 1 end exponent space open parentheses fraction numerator 16 over denominator square root of 366 end fraction close parentheses.

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Three Dimensional Geometry

Find the angle between the line $straight x over 4 space equals space fraction numerator straight y minus 2 over denominator 3 end fraction space equals space fraction numerator straight z minus 1 over denominator 6 end fraction$ and the plane x + 2y + 2 + 3 = 0

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.

Mock Test Series

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Entrance Exams Preparation

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For Daily Free Study Material Join wiredfaculty

Three Dimensional Geometry

Find the angle between the line and the plane x + 2y + 2 + 3 = 0

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.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Find the angle between the line $straight x over 4 space equals space fraction numerator straight y minus 2 over denominator 3 end fraction space equals space fraction numerator straight z minus 1 over denominator 6 end fraction$ and the plane x + 2y + 2 + 3 = 0

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