Question

Which is the vector quantity that describes the shortest path between two points?

Answer 1

Sorry, I meant displacement.

Answer 2

Mathematically, the gradient vector points in the direction of the shortest distance.

Answer 3

Is this a physics or math question?

Answer 4

That shortest path between the vectors a and b is their difference, a-b or b-a, depending on which direction you want to look. This is different than what @PsiSquared is describing, and I believe he is wrong.

While travelling on a surface, the gradient vector points in the direction of the steepest incline. Since you're travelling on the surface itself, suppose we're on the surface z=1/(x^2+y^2). The gradient will obviously point towards the center. So going from the point (1,1, 1/2) to (-1,-1, 1/2) is going to say that the shortest path is to go through the center, which is obviously wrong since we have to go through an infinite distance to get past the middle.

Answer 5

ITS DISPLACEMENT