Find the distance between the parallel planes:2x – 3y + z + 3
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    Three Dimensional Geometry

    Multiple Choice QuestionsShort 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

    The equation of given planes are
    2x – 3y + z + 3 = 0    .....(1)
    and 4x –6y + 2z + 5 = 0    .....(2)
    Let (x1.y1, z1) be any point on plane (1)
    ∴  2x1 – 3y1 + z1 + 3 = 0
    ∴  2x1 – 3y1 + z1 = –3    ...(3)
    Length of perpendicular from (x1, b 1, b1) to plane (2) in
    equals space fraction numerator open vertical bar 4 straight x subscript 1 space minus space 6 space straight y subscript 1 space plus space 2 straight z subscript 1 space plus space 5 close vertical bar over denominator square root of 16 plus 36 plus 4 end root end fraction space equals space fraction numerator open vertical bar 2 space left parenthesis 2 straight x subscript 1 space minus space 3 straight y subscript 1 space plus space straight z subscript 1 right parenthesis space plus space 5 close vertical bar over denominator square root of 56 end fraction equals space open vertical bar fraction numerator 2 left parenthesis negative 3 right parenthesis plus 5 over denominator square root of 56 end fraction close vertical bar space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space of space left parenthesis 3 right parenthesis close square brackets equals space fraction numerator 1 over denominator square root of 56 end fraction

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    257. Find the distance between parallel planes:
    2x  – y + 3z – 4 = 0 and 6x – 3y + 9z + 13 = 0.
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    258. Find the distance between the planes straight r with rightwards arrow on top. space open parentheses straight i with hat on top space plus space 2 space straight j with hat on top space plus space space 3 space straight k with hat on top close parentheses space plus space 7 space equals 0 and straight r with rightwards arrow on top. space open parentheses 2 straight i with hat on top space plus space 4 space straight j with hat on top space plus space 6 space straight k with hat on top close parentheses space plus space 7 space equals space 0.
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    Multiple Choice QuestionsLong Answer Type

    259. Find the length and coordinates of the foot of the perpendicular from the point (1, 1, 2) to the plane 2x – 2y + 4z + 5 = 0.
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    260. Find the distance between the point P(6, 5, 9) and the plane determined by the points A(3, –1, 2), B(5, 2, 4) and C(– 1, – 1, 6).
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