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Question
Solve the following :
Find the vector equation of the plane which is at a distance of 5 units from the origin and which is normal to the vector `2hat"i" + hat"j" + 2hat"k"`.
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Solution
If `hat"n"` is a unit vector along the normal and p i the length of the perpendicular from origin to the plane, then the vector equation of the plane `bar"r".hat"n" = p`
Here, `bar"n" = 2hat"i" + hat"j" + 2hat"k"` and p = 5
∴ `|bar"n"| = sqrt(2^2 + 1^2 + (2)^2`
= `sqrt(9)`
= 3
`hat"n" = bar"n"/|bar"n"|`
= `(1)/(3)(2hat"i" + hat"j" + 2hat"k")`
∴ the vector equation of the required plane is
`bar"r".[1/3(2hat"i" + hat"j" + 2hat"k")]` = 5
i.e. `bar"r".(2hat"i" + hat"j" + 2hat"k")` = 15.
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