The question came to my mind when trying to explain snowboarding up a ramp. When you don't use an edge and the snowboard is flat, it is only stable if your momentum is going straight up the ramp. You are likely to fall if your velocity is at an angle. So I was trying to explain which vector of the ramp you need to align to, but I struggled to define it.
I know that a plane can be defined with a vector that's perpendicular to it, but there are many vectors that are parallel to it. If the Z axis is gravity, I think "going straight up" means maximizing dz when stepping dx dy. I think there is only one such vector that is parallel to the plane. Is this a well defined vector? Does it have a name? I also suspect that given such a vector, you could derive its origin intersecting plane.
Suppose $Z$ is a unit vector in the vertical direction, and $N$ is a unit vector that’s normal to the plane of the hill. Then the cross product $H = N \times Z$ is the horizontal direction on the slope. In other words, if you go in the direction of $H$, you will be neither gaining nor losing altitude. The vector $N \times H$ gives you the steepest downhill direction. I think that’s what you’re asking about.
This direction is often called the “direction of steepest descent”. This terminology comes from optimization theory, where sometimes the best way to get to a minimum is to go downhill as fast as you can. The term “gradient descent” is also used.