Normal vector of velocity

Question

I'm currently reading a textbook about fluid simulations, and I keep encountering the reference to the normal of velocity. My understanding of a normal vector is that it represents a vector that is orthogonal to every other vector in the system of interest. For instance, the normal vector of $[1, 0, 0]$ and $[0, 1, 0]$ is $[0, 0, 1]$. This is, for instance, how the axis are defined.

However, in the case of velocity, I do not understand what this would represent? Is it simply talking about the above, or is there more purpose to it?

An excerpt from the textbook is as follows:

Answer 1

Remember, the dot product is a way of measuring how much a vector is pointing in a given direction. In the excerpt you provided, the condition means that no fluid is flowing into or out of the wall hence the velocity vector has no contribution from these directions so the dot product should be 0.

Answer 2

In your quoted excerpt from the textbook, the normal vector $\hat{n}$ on any given point of the surrounding surface wall is orthogonal (i.e., "normal") to the tangent plane at this surface point.

The velocity vector $\vec{u}$ at this surface point cannot have a component along the normal vector $\tilde{n}$, since the boundary wall is assumed to be impermeable for the fluid. Hence we have $\hat{n} \cdot \vec{u} = 0$.