Question

The points (2, 3,6) and (6, 3, !2) both lie on the sphere x
2
+ y
2
+ z
2
= 49. What is the shortest
path between them if travel is restricted to the surface of the sphere?

Answer 1

The second point should be -2, just so you know

Answer 2

The shortest path will be along a great circle along the surface, which will be an arc with radius 7, the radius of the sphere. The length of the arc will be 7 theta, where theta (in radians) is the angle between two lines from the center to the two points on the sphere. Calculate that angle (theta) and you will have it. You can do it with a vector cross product.

7*7 sin theta = |R1 x R2|
R1 = 2i + 3j + 6k
R2 = 6i + 3j - 2k
etc

I will leave the rest to you. Perhaps another teacher has an easier way, but this looks like a good way to do it.