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Question
Find the vector equation of a plane at a distance 6 units from the origin and to which vector `2hat"i" - hat"j" + 2hat"k"` is normal
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Solution
Let `bar"n" = 2hat"i" - hat"j" + 2hat"k"`
∴ `hat"n"` is the unit vector along normal
∴ `hat"n" = (bar"n")/|bar"n"|`
= `(2hat"i" - hat"j" + 2hat"k")/sqrt(2^2 + (-1)^2 2 ^2)`
= `(2hat"i" - hat"j" + 2hat"k")/sqrt(4 + 1 + 4)`
= `(2hat"i" - hat"j" + 2hat"k")/3`
and p = 6
Vector equation of plane is `bar"r"*hat"n"` = p
∴ `bar"r"*((2hat"i" - hat"j" + 2hat"k"))/3` = 6
∴ `bar"r"*(2hat"i" - hat"j" + 2hat"k")` = 18
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