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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.

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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 $straight r with rightwards arrow on top. space open parentheses 2 straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top close parentheses space equals space 4$

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256. Find the distance between the parallel planes:
2x – 3y + z + 3 = 0
and 4x – 6y + 2z + 5 = 0

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257. Find the distance between parallel planes:
2x – y + 3z – 4 = 0 and 6x – 3y + 9z + 13 = 0.

The equations of given planes are
2x – y + 3 z – 4 = 0        ... (1)
and 6 x – 3 y + 9 z + 13 = 0        ....(2)
Let (x₁, y₁ , z₁ ) be any point on plane (1)
∴ 2 x₁ –y₁ + 3 z₁ – 4 = 0 or 2x₁– y, + 3 z₁ =4    . (3)
Length of perpendicular from (x₁, y₁ , z₁ ) to the plane (2) is
$equals space fraction numerator open vertical bar 6 straight x subscript 1 minus 3 straight y subscript 1 plus 9 straight z subscript 1 plus 13 close vertical bar over denominator square root of left parenthesis 6 right parenthesis squared plus left parenthesis negative 3 right parenthesis squared plus left parenthesis 9 right parenthesis squared end root end fraction space equals space fraction numerator open vertical bar 3 space open parentheses 2 straight x subscript 1 space minus space straight y subscript 1 space plus space 3 space straight z subscript 1 close parentheses plus 13 close vertical bar over denominator square root of 36 plus 9 plus 81 end root end fraction space equals fraction numerator open vertical bar 3 space left parenthesis 4 right parenthesis space plus space 13 close vertical bar over denominator square root of 126 end fraction space equals fraction numerator 25 over denominator square root of 126 end fraction$
which is required distance between the planes.