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Question
Find the vector and Cartesian equation of the plane passing through the point with position vector `2hat"i" + 6hat"j" + 3hat"k"` and normal to the vector `hat"i" + 3hat"j" + 5hat"k"`
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Solution
Given `vec"a" = 2hat"i" + 6hat"j" + 3hat"k"` and `vec"n" = hat"i" + 3hat"j" + 5hat"k"`
Vector equation of the plane
`vec"r"*vec"n" = vec"a"*vec"n"`
`vec"r"*vec"n" = (2hat"i" + 6hat"j" + 3hat"k")(hat"i" + 3hat"j" + 5hat"k")`
`vec"r"*(vec"i" + 3vec"j" + 5vec"k")` = 12 + 8 + 15
⇒ `vec"r"*(vec"i" + 3vec"j" + 5vec"k")` = 35
Cartesian equation of the plane
`(xvec"i" + yvec"j" + zvec"k")*(vec"i" + 3vec"j" + 5vec"k")` = 35
x + 3y + 5z = 35
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