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What I did was use the distance formula from the plane to the origin, found the normal vector, and found the unit vector of the normal vector. I multiplied the unit vector by the distance to get the point (and it satisfies the equation). Is this correct?

Is there a way of doing this using optimization (i.e. local min/max)?

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Yes, you can. It will be more work that what you did. – Ross Millikan Apr 9 '13 at 1:39
Or you can just notice that the normal is $(2, -1, 2)$ and find where the line $(2, -1, 2)t$ intersects your plane. – user66345 Apr 9 '13 at 1:42
up vote 2 down vote accepted

Sounds right.

Yes, you can do this by minimization. First write down the distance of a general point from the origin (hint: it is simple). Then you want to minimize that function subject to the constraint given by the plane.

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