I know how to compute the unit vector
$$\hat{\textbf{u}} = \left( \frac{u_1}{||u||} , \dots , \frac{u_n}{||u||} \right)$$
and I also know how to show that this will have length 1 by using the definition of a norm.
What I do not understand is why dividing each coordinate by a scalar (in this case the norm of u) maintains the direction of the original vector.
Is there some axiom that says that dividing/multiplying each coordinate by a scalar always maintains direction? Is there a way prove the direction does not change?
It's easy to see how things scale up and down in 2D, but I'm curious if there is a way to show that the direction is always maintained regardless of the dimension.
Two nonzero vectors $a, b$ are parallel iff $a=\lambda b$ for some $\lambda\in\Bbb R$, and if this $\lambda$ is positive, they point to the same direction.