Why does gradient ascent/descent exhibit zig-zag motion?

Question

A good way to visualize gradient ascent/descent is to assume you are in a quadratic bowl or on a mountain. If I visualize this, then the direction of steepest ascent/descent is the one that points straight towards the bottom of the bowl or top of the mountain.

With this understanding I have two questions:

1. If you want to climb the hill or go down a bowl, why take a zig-zag path instead of taking a straight path to the top/bottom?

2. Why doesn't the steepest path have a unit in the z direction? I understand that gradient is orthogonal to the level sets of the function. That is it lies in the x-y plane orthogonal to the contour. But why doesn't it have a unit in the z direction? With a unit in the z direction, it can point towards the minima/maxima and still be orthogonal to contour lines.

I have a related question: Gradient is NOT the direction that points to the minimum or maximum

Answer 1

Here's the plot of the gradient of a certain bowl. Notice that while the arrows point "uphill", they don't necessarily point directly away from the minimum at $(0,0)$.

The straight path to the bottom will not, in general, follow the direction of steepest descent.