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Question
The foot of the perpendicular drawn from the origin to a plane is M(1, 2, 0). Find the vector equation of the plane.
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Solution
The vector equation of the plane passing through `A(bara)` and perpendicular to `barn` is `barr.barn = bara.barn`.
M(1, 2, 0) is the foot of the perpendicular drawn from the origin to the plane.
Then the plane is passing through M and is perpendicular to OM.
If `barm` is the position vector of M, then `barm = hati + 2hatj`.
Normal to the plane is `barn = bar(OM) = hati + 2hatj`
`barm.barn = (hati + 2hatj) . (hati + 2hatj)`
= 1(1) + 2(2)
= 1 + 4
= 5
∴ The vector equation of the required plane is `barr.(hati + 2hatj)` = 5.
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