Distance between a point and a surface (sphere)

[JustAChemist]

Homework Statement

What function represents the distance between the point P(1, 2, 2) and any point on the sphere
x^2 + y^2 + z^2 = 1?

Homework Equations

My solutions manual says that the answer is f = (x - 1)^2 + (y - 2)^2 + (z - 2)^2

The Attempt at a Solution

I tried applying pythagoras' theorem and came up with f = sqrt[ (1 - x)^2 + (2 - y)^2 + (2 - z)^2 ] :/

It's actually part of an optimisation problem (what are the shortest and longest distances from P and the sphere?), but I'm pretty sure I've got it sorted once I can get the function... can anyone help?... please? ): (without vectors if possible :/)

thankyou (:

 

[Hurkyl]

Who cares if you do the work differently than the solution manual? There are lots of ways to do problems, and it's usually best to do things in a way you understand rather than trying to mimic exactly something you don't understand.

And sometimes solutions manuals are flat-out wrong.

(But after the fact, it is often worth studying something you don't understand to learn why it works and how it compares to what you do understand)


Anyways, your function looks like it really is the distance formula. I imagine the method of the solution manual and the method you intend to use are similar, with the only difference is that they simplified the problem early on, so that their later work will be easier.

 

[JustAChemist]

Thankyou (:

> The solutions manual, however, used an algebraic trick that is commonly useful.

I apologise for asking, but would you mind sharing the trick with me? (:

 

[Hurkyl]

Well, you do recognize the similarities between the two functions you wrote? I bet part of the work in solutions manual is to find the optima of that function?

 

[JustAChemist]

yeah it was. I got those correctly using the function I 'borrowed' from the solution manual, but I didn't use mine because I assumed it was wrong lol :P

thanks a lot for the help (:

embarassingly, I had been stuck with this for a couple hours lol... :/

 

[HallsofIvy]

The simplest way to do this is to write out the equation of the line through the given point and the center of the sphere. The points at which that line crosses the sphere will be the points on the sphere nearest to and farthest from the given point.

The sphere given by x^2 +y^2+ z^2+ 1 has center at (0, 0, 0) and the line through (0, 0, 0) and (1, 2, 2) is given by x= t, y= 2t, z= 2t.

The points where that line crosses the sphere must satisfy (t)^2+ (2t)^2+ (2t)^2= 1 or 9t^2= 1 so that t= 1/3 or t= -1/3.

Since (1, 2, 2) has t= 1, the point with t= 1/3, (1/3, 2/3, 2/3), is the point on the sphere nearest to (1, 2, 2) and the point with t= -1/3, (-1/3, -2/3, -2/3) is the point on the sphere farthest from (1, 2, 2).