Find the equation of the plane passing through the points (3, 2,

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Three Dimensional Geometry

Long Answer Type

281. Find the vector and cartesian equation of the plane containing the two lines

straight r with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space straight j with hat on top space minus space 3 space straight k with hat on top space plus space straight lambda space left parenthesis straight i with hat on top space plus space 2 space straight j with hat on top space plus space 5 space straight k with hat on top right parenthesis

and

straight r with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space space 3 space straight j with hat on top space minus space 7 space straight k with hat on top space plus space straight mu space left parenthesis 3 straight i with hat on top space minus space 2 space straight j with hat on top space plus space space 5 space straight k with hat on top right parenthesis

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Short Answer Type

282. Find the equation of the plane containing the lines:

fraction numerator straight x minus 4 over denominator 1 end fraction space equals space fraction numerator straight y minus 3 over denominator 4 end fraction space equals space fraction numerator straight z minus 2 over denominator 5 end fraction

and

fraction numerator straight x minus 3 over denominator 1 end fraction space equals space fraction numerator straight y minus 2 over denominator negative 4 end fraction space equals space straight z over 5

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Long Answer Type

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283. Find the equation of the plane passing through the points (3, 2, 1) and (0, 1, 7) and parallel to the line $straight r with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top space plus space straight lambda left parenthesis straight i with hat on top space minus space straight j with hat on top space minus space straight k with hat on top right parenthesis.$

The given points are (3, 2, 1) and (0, 1, 7)

therefore space space space space straight a with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space 2 space straight j with hat on top space plus space straight k with hat on top comma space space straight b with rightwards arrow on top space equals space straight j with hat on top space plus space 7 space straight k with hat on top

The equation of line is

straight r with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top space space plus space straight lambda space left parenthesis straight i with hat on top space minus space straight j with hat on top space minus space straight k with hat on top right parenthesis

therefore space space space space straight c with rightwards arrow on top space equals space straight i with hat on top space minus space straight j with hat on top space minus space straight k with hat on top

therefore space space space straight b with rightwards arrow on top space cross times space straight c with rightwards arrow on top space equals space open vertical bar table row cell straight i with hat on top end cell cell straight j with hat on top end cell cell straight k with hat on top end cell row 0 1 7 row 1 cell negative 1 end cell cell negative 1 end cell end table close vertical bar space equals space straight i with hat on top space open vertical bar table row 1 7 row cell negative 1 end cell cell negative 1 end cell end table close vertical bar space minus space straight j with hat on top space open vertical bar table row 0 7 row 1 cell negative 1 end cell end table close vertical bar space plus space straight k with hat on top open vertical bar table row 0 1 row 1 cell negative 1 end cell end table close vertical bar space space space space space space space space space space space space space space space space space space space space equals space left parenthesis negative 1 plus 7 space right parenthesis straight i with hat on top space minus space left parenthesis 0 minus 7 right parenthesis space straight j with hat on top space plus space left parenthesis 0 minus 1 right parenthesis space straight k with hat on top space equals space 6 space straight i with hat on top space plus space 7 space straight j with hat on top space minus space straight k with hat on top

straight c with rightwards arrow on top cross times straight a with rightwards arrow on top space equals space open vertical bar table row cell straight i with hat on top end cell cell straight j with hat on top end cell cell straight k with hat on top end cell row 1 cell negative 1 end cell cell negative 1 end cell row 3 2 1 end table close vertical bar space equals space straight i with hat on top space open vertical bar table row cell negative 1 end cell cell negative 1 end cell row 2 1 end table close vertical bar space minus space straight j with hat on top space open vertical bar table row 1 cell negative 1 end cell row 3 1 end table close vertical bar space plus space straight k with hat on top open vertical bar table row 1 cell negative 1 end cell row 3 2 end table close vertical bar

equals left parenthesis negative 1 plus 2 right parenthesis space straight i with hat on top space minus space left parenthesis 1 plus 3 right parenthesis space straight j with hat on top space plus space left parenthesis 2 plus 3 right parenthesis space straight k with hat on top space equals space straight i with hat on top space minus space 4 space straight j with hat on top space plus space 5 space straight k with hat on top

therefore space space open square brackets straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top close square brackets space equals space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space cross times space straight c with rightwards arrow on top space equals space open parentheses 3 space straight i with hat on top space plus space 2 space straight j with hat on top space plus space straight k with hat on top close parentheses space. space open parentheses 6 space straight i with hat on top space plus space 7 space straight j with hat on top space minus space straight k with hat on top close parentheses

equals space left parenthesis 3 right parenthesis thin space left parenthesis 6 right parenthesis space plus space left parenthesis 2 right parenthesis thin space left parenthesis 7 right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis negative 1 right parenthesis space equals space 18 plus 14 minus 1 space equals space 31

Required equation of plane is

open square brackets straight r with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top close square brackets space plus space open square brackets straight r with rightwards arrow on top comma space straight c with rightwards arrow on top comma space straight a with rightwards arrow on top close square brackets space equals space open square brackets straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top close square brackets

or space space straight r with rightwards arrow on top. space open parentheses straight b with rightwards arrow on top space cross times space straight c with rightwards arrow on top close parentheses space plus space straight r with rightwards arrow on top. space open parentheses straight c with rightwards arrow on top space cross times space straight a with rightwards arrow on top close parentheses space equals space open square brackets straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top close square brackets or space space straight r with rightwards arrow on top. space open curly brackets open parentheses straight b with rightwards arrow on top cross times straight c with rightwards arrow on top close parentheses space plus space open parentheses straight c with rightwards arrow on top cross times straight a with rightwards arrow on top close parentheses close curly brackets space equals space open square brackets straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top close square brackets or space space space straight r with rightwards arrow on top. space open parentheses 6 space straight i with hat on top plus 7 stack space straight j with hat on top space minus space straight k with hat on top space plus space straight i with hat on top space minus space 4 space straight j with hat on top space plus space 5 space straight k with hat on top close parentheses space equals space 31 or space space straight r with rightwards arrow on top. space open parentheses 7 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 4 space straight k with hat on top close parentheses space equals space 31.

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284. Find the equation of the plane passing through the points (1, 2, 3) and (0. –1, 0) and parallel to the line

fraction numerator straight x minus 1 over denominator 2 end fraction space equals space fraction numerator straight y plus 2 over denominator 3 end fraction space equals space fraction numerator straight z over denominator negative 3 end fraction.

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285. Find the equation of the plane passing through the points (3, 2, 2) and (1, 0, –1) parallel to the line

fraction numerator straight x minus 1 over denominator 2 end fraction space equals space fraction numerator straight y minus 1 over denominator negative 2 end fraction space equals fraction numerator straight z minus 2 over denominator 3 end fraction.

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286. Find the equation of the plane passing through the origin and parallel to the vectors

straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top space space space and space space 3 space straight i with hat on top space minus space straight k with hat on top.

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Short Answer Type

287. Find the vector and Cartesian form of the equation of the plane passing through the point (1, 2, –4) and parallel to the lines

straight r with rightwards arrow on top space equals straight i with hat on top space plus space 2 space straight j with hat on top space minus space 4 space straight k with hat on top space plus space straight lambda open parentheses 2 straight i with hat on top space plus space 3 space straight j with hat on top space plus space 6 space straight k with hat on top close parentheses

and

straight r with rightwards arrow on top space equals space straight i with hat on top space minus space 3 space straight j with hat on top space plus space space 5 space straight k with hat on top space plus space straight mu space open parentheses 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

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Long Answer Type

288. Show that the lines:

fraction numerator straight x minus 1 over denominator 2 end fraction space equals fraction numerator straight y minus 2 over denominator 3 end fraction space equals space fraction numerator straight z minus 3 over denominator 4 end fraction space and space fraction numerator straight x minus 4 over denominator 5 end fraction space equals space fraction numerator straight y minus 1 over denominator 2 end fraction space equals straight z

intersect. Find the point of intersection also.

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289. Show that the lines

fraction numerator straight x plus 3 over denominator negative 3 end fraction space equals space fraction numerator straight y minus 1 over denominator 1 end fraction space equals space fraction numerator straight z minus 5 over denominator 5 end fraction space and space fraction numerator straight x plus 1 over denominator negative 1 end fraction space equals fraction numerator straight y minus 2 over denominator 2 end fraction space equals fraction numerator straight z minus 5 over denominator 5 end fraction

are coplanar.

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290. Show that the lines

fraction numerator straight x minus 5 over denominator 4 end fraction space equals space fraction numerator straight y minus 7 over denominator 4 end fraction space equals space fraction numerator straight z plus 3 over denominator negative 5 end fraction

and

fraction numerator straight x minus 8 over denominator 7 end fraction space equals space fraction numerator straight y minus 4 over denominator 1 end fraction space equals space fraction numerator straight z minus 5 over denominator 3 end fraction

intersect each other.

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