Find the distance between the point P(6, 5, 9) and the plane det

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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.

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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).

The equation of plane through the points (3, –1, 2), (5, 2, 4), (– 1, – 1, 6) is

open vertical bar table row cell straight x minus 3 end cell cell straight y plus 1 end cell cell straight z minus 2 end cell row cell 5 minus 3 end cell cell 2 plus 1 end cell cell 4 minus 2 end cell row cell negative 1 minus 3 end cell cell negative 1 plus 1 end cell cell 6 minus 2 end cell end table close vertical bar space equals 0 space space space space space space space space space space space open square brackets open vertical bar table row cell straight x minus straight x subscript 1 end cell cell straight y minus straight y subscript 1 end cell cell straight z minus straight z subscript 1 end cell row cell straight x subscript 2 minus straight x subscript 1 end cell cell straight y subscript 2 minus straight y subscript 1 end cell cell straight z subscript 2 minus straight z subscript 1 end cell row cell straight x subscript 3 minus straight x subscript 1 end cell cell straight y subscript 3 minus straight y subscript 1 end cell cell straight z subscript 3 minus straight z subscript 1 end cell end table close vertical bar space equals space 0 close square brackets

or

open vertical bar table row cell straight x minus 3 end cell cell straight y plus 1 end cell cell straight z minus 2 end cell row 2 3 2 row cell negative 4 end cell 0 4 end table close vertical bar space equals space 0

or space space space space left parenthesis straight x minus 3 right parenthesis space open vertical bar table row 3 2 row 0 4 end table close vertical bar space minus space left parenthesis straight y plus 1 right parenthesis space open vertical bar table row 2 2 row cell negative 4 end cell 4 end table close vertical bar space plus space left parenthesis straight z minus 2 right parenthesis space open vertical bar table row 2 3 row cell negative 4 end cell 0 end table close vertical bar space equals space 0

or space space space space space space left parenthesis straight x minus 3 right parenthesis thin space left parenthesis 12 minus 0 right parenthesis space minus space left parenthesis straight y plus 1 right parenthesis space left parenthesis 8 plus 8 right parenthesis space plus space left parenthesis straight z minus 2 right parenthesis space left parenthesis 0 plus 12 right parenthesis space equals space 0 or space space space space space space 12 space left parenthesis straight x minus 3 right parenthesis space minus space 16 space left parenthesis straight y plus 1 right parenthesis space plus space 12 space left parenthesis straight z minus 2 right parenthesis space equals space 0 or space space space space space space 3 left parenthesis straight x minus 3 right parenthesis space space minus space 4 space left parenthesis straight y plus 1 right parenthesis space plus space 3 space left parenthesis straight z minus 2 right parenthesis space equals space 0 or space space space space space space 3 straight x minus 9 minus 4 straight y minus 4 plus 3 straight z minus 6 space equals space 0 or space space space space space 3 straight x minus 4 straight y plus 3 straight z minus 19 space equals space 0

Let p be the perpendicular distance of P(6, 5, 9) from this plane.

therefore space space space space space space straight p space equals space fraction numerator 3 left parenthesis 6 right parenthesis minus 4 left parenthesis 5 right parenthesis space plus space 3 left parenthesis 9 right parenthesis space minus space 19 over denominator square root of 9 plus 16 plus 9 end root end fraction space equals fraction numerator 18 minus 20 plus 27 minus 19 over denominator square root of 34 end fraction equals space fraction numerator 6 over denominator square root of 34 end fraction

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