Read up your textbook or lecture notes on Lagrange multipliers. It is not as difficult as it sounds.
However, you need to master you algebra and basic calculus, which I believe should not be a problem.
To put it in the least words possible, we attempt to optimize (maximize or minimize) a function f(x,y,z) subject to the constraint g(x,y,z).
The suggested objective function is the square of the distance from the origin, which therefore is
The constraint is
We now introduce a Lagrange multiplier, λ, to form a new function Λ:
Now apply partial differentiation with respect to each of the variables, and equate result to zero:
Add the original constraint equation:
Now solve the system of 4 equations in x,y,z and λ and voilą!
x=1, y=-2, z=3, L=2.
So the distance is
How can we tell if this is correct?
This is not too difficult... in this particular case.
The shortest distance from a point (origin 0,0,0) to a plane is the perpendicular distance, given by the well-known formula:
and the square of the minimum distance
which checks with our Lagrange multiplier answer.
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