Shortest distance between two points in three dimensional

Question

Question) show that the shortest distance between two points in three dimensional space is a straight line?

Question) Finding the shortest path between any two points on the surface of a sphere but use the method of the Euler equations with an auxiliary condition imposed?

Question) Find the shortest path between the (x,y,z) points (0,-1,0) and (0,1,0) on the conical surface z=1-(x²+y²)^1/2 what is the length of the path?

Question)

(a) Find the curve on the surface z=x^3/2 joining the points(x,y,z)=(0,0,0) and (1,1,1) has the shortest arc length?

(b) Use a computer to produce a plot showing the surface and the shortest curve on a single plot?