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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.

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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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Long 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.

The equation of plane is
2x – 2y + 4 z + 5 = 0 ...(1)
Direction ratios of the normal to the plane are
2,–2, 4 i.e., 1,– 1, 2.
Let M be foot of perpendicular from P (1, 1, 2) to the plane.
Now PM is a straight line which passes through P (1. 1,2) and has direction ratios as 1, – 1, 2.
∴ its equations are

$fraction numerator straight x minus 1 over denominator 1 end fraction space equals space fraction numerator straight y minus 1 over denominator negative 1 end fraction space equals space fraction numerator straight z minus 2 over denominator 2 end fraction space equals space straight r space left parenthesis say right parenthesis$
Any point M on line is (r + 1, – r + 1, 2 r + 2)
∵ M lies on plane (1)
∴ 2 (r + 1) – 2 (– r + 1) + 4 (2 r + 2) + 5 = 0.
∴ 2r + 2 + 2r – 2 + 8r + 8 + 5 = 0
$rightwards double arrow space space space space 12 space straight r space equals space minus 13 space space space space rightwards double arrow space space space space straight r space equals space minus 13 over 12$
$therefore space space space space space straight M space is space space open parentheses negative 13 over 12 plus 1 comma space space 13 over 12 plus 1 comma space space minus 26 over 12 plus 2 close parentheses space space or space space space open parentheses negative 1 over 12 comma space space 25 over 12 comma space minus 2 over 12 close parentheses therefore space space space space space foot space of space perpendicular space is space open parentheses negative 1 over 12 comma space 25 over 12 comma space minus 1 over 6 close parentheses Length space of space perpendicular space space equals space fraction numerator 2 left parenthesis 1 right parenthesis space minus space 2 left parenthesis 1 right parenthesis space plus space 4 left parenthesis 2 right parenthesis space plus space 5 over denominator square root of left parenthesis 2 right parenthesis squared plus left parenthesis negative 2 right parenthesis squared plus left parenthesis 4 right parenthesis squared end root end fraction space equals fraction numerator 2 minus 2 plus 8 plus 5 over denominator square root of 4 plus 4 plus 16 end root end fraction space equals fraction numerator 13 over denominator square root of 24 end fraction$

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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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