Originally Posted by

**Soroban** Hello, yungman!

I think your questions are asking for *undefined* quantities.

$\displaystyle \hat A \:=\:\dfrac{\vec A}{|\vec A|}\,\text{ is a unit vector in the direction of }\vec A. $

If $\displaystyle |\vec A| = 0$, the vector has length 0; it is a point, $\displaystyle \bullet$

. . It has *no** *__direction__.

The unit vector does not exist.

Now you are asking for the magnitude of a unit vector that does not exist.

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Having said that, I did a "mind experiment."

We have a vector: .$\displaystyle \vec v \:=\:\langle a,b\rangle$

Its unit vector is:. $\displaystyle \vec u \:=\:\dfrac{\vec v}{|\vec v|} \;=\;\dfrac{\langle a,b\rangle}{\sqrt{a^2+b^2}} $

And you claim that: .$\displaystyle \displaystyle \lim_{a,b\to0} |\vec u| \;=\;\lim_{a,b\to0}\left|\frac{\langle a,b\rangle}{\sqrt{a^2+b^2}}\right| \;=\;1 $

That is, that *all* vectors (however small) have a unit vector of length 1.

I must admit that your claim has merit . . .