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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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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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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.

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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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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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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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