The minimal distance between any point on the sphere1 $\displaystyle (x-2)^{2}+(y-1)^{2}+(z-3)^{2}=1$ and $\displaystyle (x+3)^{2}+(y-2)^{2}+(z-4)^{2}=4$ (sphere2) is...?

Looking at the range of values

$\displaystyle 1 \leq x \leq 3,

0 \leq y \leq 2,

2 \leq z \leq 4$ for sphere1 and

$\displaystyle -5 \leq x \leq -1,

0 \leq y \leq 4,

2 \leq z \leq 6$ for the sphere2, I decided that the minimal distance would occur at x=1 for sphere1 and x=-1 for sphere2 because these are the x-coordinates that are closest while still being on their respective spheres.

If I plug those in, it gives me

sphere1: $\displaystyle (y-1)^{2}+(z-3)^{2}=0$ and

sphere2: $\displaystyle (y-2)^{2}+(z-4)^{2}=0$. I think I should be doing something with these equations but I'm not sure what that would be.

I also decided that the minimal distance must lie along the line connecting the centers of the circles, so I used (2,1,3) - (-3,2,4) = <5,-1,-1> as a position vector and the coordinates of the center of sphere1 to give me an equation of a line: (2,1,3) + t<5,-1,-1>.

This is where I'm stuck. Using x=1 and then x=-1, I solved x=2+5t for t, then plugged these values back into the equations y =1-t and z=3-t to get the y and z coordinates, but the distance between the two points is incorrect (I have the correct answer; it's $\displaystyle 3(\sqrt{3}-1)$. I calculated $\displaystyle (1, \frac{6}{5},\frac{16}{5})$ and $\displaystyle (-1, \frac{8}{5},\frac{18}{5})$ for the points, and the distance between them is $\displaystyle \frac{6}{5}\sqrt{3}$.