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Question
Find the non-parametric form of vector equation and cartesian equation of the plane passing through the point (1, − 2, 4) and perpendicular to the plane x + 2y − 3z = 11 and parallel to the line `(x + 7)/3 = (y + 3)/(-1) = z/1`
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Solution
The required plane passing through the point `vec"a" = vec"i" - 2vec"j" + 4vec"k"` and parallel to the plane `vec"b" = vec"i" + 2vec"j" - 3vec"k"` and parallel to the line `vec"c" = 3vec"i" - vec"j" + vec"k"`
`vec"b" xx vec"c" = |(vec"i", vec"j", vec"k"),(1, 2, -3),(3, -1, 1)|`
= `vec"i"(2 - 3) - vec"j"(1 + 9) + vec"k"(-1 - 6)`
`vec"b" xx vec"c" = -vec"i" - 10vec"j" - 7vec"k"`
Non-parametric form of vector equation
`(vec"r" - vec"a")*(vec"b" xx vec"c")` = 0
`(vec"r" - vec"a")*(-vec"i" - 10vec"j" - 7vec"k")` = 0
`vec"r"*(-vec"i" - 10vec"j" - 7vec"k") = vec"a"*(-vec"i" - 10vec"j" - 7vec"k")`
`vec"r"*(-vec"i" - 10vec"j" - 7vec"k") = (vec"i" - 2vec"j" + 4vec"k")(-vec"i" - 10vec"j" - 7vec"k")`
`vec"r"*(-vec"i" - 10vec"j" - 7vec"k")` = – 1 + 20 – 28
`vec"r"*(-vec"i" - 10vec"j" - 7vec"k")` = – 9
or
`vec"r"*(vec"i" + 10vec"j" + 7vec"k")` = 9
Cartesian equation
`(xvec"i" + yvec"j" + zvec"k")*(vec"i" + 10vec"j" + 7vec"k")` = 9
x + 10y + 7z – 9 = 0
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