Question

Find the point on the sphere x2+y2+z2=9 that is closest to the point (2,3,4).
What theory should I used for this question? Can I use extreme value theorem?

Answer 1

I recommend you consider using the DISTANCE FORMULA. After all, you're trying to minimize a distance.

Please note that x2+y2+z2=3^2=9 would be much clearer if written x^2+y^2+z^2=3^2.

Answer 2

Note that this represents a sphere of radius 3. You might want to sketch this sphere and then locate the point (2,3,4). Would this point be located inside the sphere or outside the sphere? You are familiar with minimizing functions of 1 variable. Can you recall discussion in your textbook of minimizing functions of 2 or 3 variables?

Answer 3

oh okay thank you so much!

Answer 4

that point lies at a distance \(\sqrt{2^2 + 3^2 + 4^2} = \sqrt{29}\) from the origin, so outside the sphere, which has radius 3 !


the line of shortest distance from the point to the sphere will hit the sphere at a vector that is normal to the sphere.

any point on the sphere, given by \(\mathbf r = <x,y,z>\) will have a normal vector \(\mathbf n = \nabla (x^2+y^2+z^2) \equiv <x,y,z>\). And that \(\mathbf{n}\) vector, due not least to the symmetry, must pass through the Origin. And so the line of shortest distance between the point and sphere must also lie on that line.

So the distance is \(\sqrt{29} - 3\)