For our classes we use a very old book that doesn't have any examples and my professor isn't making it any any easier for me to understand.
So my question is about the shortest distance from a point (1,-1,1) to the set of points given by z=xy.
Now I did my research online but I can't really find anything that would fully explain to me how I am supposed to do this step by step.
And no I don't have any solution since I'm not sure how to begin.
Best Answer
Here's a starting point: suppose that point $P = (x, y, z)$ lies on the surface. Then you know that $z = xy$, so the point is really just $(x, y, xy)$. And every point on the sruface has this form for some $x$ and $y$.
The distance from $(1, -1, 1)$ to $P$ is minimized when the SQUARED distance is minimized (this just simplifies the algebra a little). That squared distance is
$$ s(x, y) = (1 - x)^2 + (-1 - y)^2 + (1 - xy)^2. $$
Now you have a function on the $xy$ plane, namely $s$, that you need to minimize. Go for it!