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Question
Find the parametric form of vector equation, and Cartesian equations of the plane passing through the points (2, 2, 1), (9, 3, 6) and perpendicular to the plane 2x + 6y + 6z = 9
Solution
The required plane passes through the points
`vec"a" = 2vec"i" + 2vec"j" + vec"k"`
`vec"b" = 9vec"i" + 3vec"j" + 6vec"k"`
And parallel to the vector `vec"c" = 2vec"i" + 6vec"j" + 6vec"k"`
`vec"b" - vec"a" = (9vec"i" + 3vec"j" + 6vec"k") - (2vec"i" + 2vec"j" + vec"k")`
= `4vec"i" + vec"j" + 5vec"k"`
Nom-aprametric form of vector equation
`(vec"r" - vec"a")*((vec"b" - vec"a") xx vec"c")` = 0
`(vec"b" - vec"a") xx vec"c" = |(vec"i", vec"j", vec"k"),(7, 1, 5),(, 6, 6)|`
= `vec"i"(6 - 30) - vec"j"(42 - 10) + vec"k"(42 - 2)`
= `- 24hat"i" - 32hat"j" + 40hat"k"`
(1) ⇒ `(vec"r" - vec"a")*(-24hat"i" - 32hat"j" + 40hat"k")` = 0
`vec"r"*(-24hat"i" - 32hat"j" + 40hat"k") = vec"a"*(-24vec"i" - 32vec"j" + 40vec"k")`
`"r"*(24vec"i" - 32vec"j" + 40vec"k") = (2vec"i" + 2vec"j" + vec"k")(-24vec"i" - 32vec"j" + 40vec"k")`
`vec"r"*(-24hat"i" - 32hat"j" + 40hat"k") = - 48 - 64 + 40`
`-8[vec"r"*(3hat"i" + 4hat"j" - 5hat"k")]` = – 72
`vec"r"*(3hat"i" + 4hat"j" - 5hat"k")` = 9
Cartesian equation
`(xvec"i" + yvec"j" + xvec"k")*(3vec"i" + 4vec"j" - 5vec"k")` = 9
`3x + 4y - 5z - 9` = 0
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