How are graphs of vector fields plotted in 3D (or 2D)?

Question

$$ \vec F =(x^2-yz ,\, y^2-zx ,\, z^2x) $$

Could such a vector field be representative of a physical phenomena? And how are we suppose to plot it out? Would also like to see a picture of this field if possible.

Answer 1

In regards to plotting this, one approach would be to compute the value of $\vec{F}$ at a number of different grid points (x,y,z). Each component of the vector would then be represented by the length of the vector in that coordinate. For example, at the point (1,2,0), the vector field would have components $(1^2-2*0, 2^2-0*1, 0^2*1)=(1,4,0)$. etc. You would repeat this for some grid of points in 2D or 3D space, depending on your needs. In regards to whether this represents a physical phenomenon, I can't say yes or no. I can't think of a reason why it would be excluded, but I don't have any examples to illustrate. I hope this helps.