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Three Dimensional Geometry

Short Answer Type

251. Find the angle between the line

fraction numerator straight x minus 1 over denominator 1 end fraction space equals space fraction numerator straight y minus 2 over denominator negative 1 end fraction space equals space fraction numerator straight z plus 1 over denominator 1 end fraction

and the plane 2x – y + z= 1.

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252.

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

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253. 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.

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254. 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.

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

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255.

Find the angle between the line $straight r with rightwards arrow on top space equals space open parentheses straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top close parentheses space plus space straight lambda space open parentheses 2 straight i with hat on top space minus space straight j with hat on top space plus space 3 space straight k with hat on top close parentheses$ and the plane $straight r with rightwards arrow on top. space open parentheses 2 straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top close parentheses space equals space 4$