Why can $v_x,v_y$ in ${\bf{v}}=(z,z,z)$ contain the $z$ variable?

Question

I am rather confused why the vector field:

$${\bf{v}}=(z,z,z)$$

can contain the $z$ variable in the $x$ and $y$ components any help with explaining this would be great.

Answer 1

A vector field assigns to every point in space a vector. This one takes a point in space and assigns the vector that is the z coordinate in all 3 directions. I would find it more natural for it to use angle brackets to indicate that the output is a vector and not a point, so

$$v(x,y,z)=\langle z,z,z \rangle$$

For example, this would assign the vector $\langle 3,3,3 \rangle$ to every point on the plane $z=3$

Answer 2

The velocities $v_{x}$ and $v_{y}$ are velocities that point in the $x$ and $y$ directions. They can also depend on depth, $z$. Think of velocities in a stream that are zero at the bottom of the river, and peak at the surface of the river. These components increase as the coordinate system elevation above the river bottom increases from 0 to $z=H$.

Alternatively, think of a head wind to a horizontally flying aeroplane. That velocity depends on elevation above ground, but points (let's say) due North.