Surfaces and dot products

[Noxide]

A little clarification is required for the following techniqueTHE TECHNIQUE
Given a surface z = f(x,y), and some point Q in R³ (not on the surface)
The point P on the surface for which the distance from P(x, y, f(x,y)) to Q is the shortest distance from the surface to Q (i.e. vector PQ has minimal length) is determined by minimizing the squared length (or PQ dot PQ)of vector PQ.

A REMARK
The next paragraph seems dauntingly long, I think it asks 2 questions...

THE QUESTIONS
That's fine and dandy as techniques go, but I'm having trouble understanding exactly why we do that to the squared length. I can understand wanting to minimize the length... but minimizing the dot product/squared length seems foreign to me. Clearly there's some gap in my knowledge as to why this is done. Also, we are finding the minimum of the new function g(x,y) = PQ dot PQ by setting it's first order partial derivatives w.r.t x and y equal to zero, but we then substitute those same values of x and y into the surface f(x,y)... I understand that x and y carry through, but it just seems odd that the values of x and y for which g(x,y) is a minimum (i'm not sure if that's always the case, but the solution manual seems to indicate it is) will yield a point P whose length is minimal to Q.

 

[tiny-tim]

Hi Noxide!

… I'm having trouble understanding exactly why we do that to the squared length. I can understand wanting to minimize the length... but minimizing the dot product/squared length seems foreign to me.

The length of PQ is defined as √(PQ.PQ).

Call that r … it doesn't matter whether we minimise (positive) r or r², they'll be at minimum or maximum together …

i] this is obvious!
ii] alternatively, if ∂r/∂x = 0, then ∂r²/∂x = 2r∂r/∂x = 0 …

and we choose to do it to r² because that avoids using square-roots, so it's quicker and easier!

(sorry, but I don't understand your second question )