The minimal distance between any point on the sphere1 and (sphere2) is...?
Looking at the range of values
for sphere1 and
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
sphere2: . 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 . I calculated and for the points, and the distance between them is .