I'm going through a textbook on GPS right now and I came across a derivation where I don't understand one of the steps. The expression
$$\left \| \widetilde{v_1} - \widetilde{v_2} - \widetilde{v_3} \right \| - \left \| \widetilde{v_1} - \widetilde{v_2} \right \| $$ is being approximated as the following:
$$ -\frac{\widetilde{v_1}-\widetilde{v_2}}{\left \|\widetilde{v_1} -\widetilde{v_2} \right \|}\cdot \widetilde{v_3} $$
where $\widetilde{v}$ represents a vector and $ \left \| \right \| $ represents the magnitude of a vector. The only note in the textbook about this derivation is "...where we have used a Taylor series approximation of a vector norm."
The only thing I could find online was this post on stack exchange and I'm not sure it's the same as this problem. Is there an online derivation of this vector normalization approximation or could anyone offer an explanation on why this is a valid approximation? Am I just missing something simple?
This comes from the Taylor approximation about $x=0$ of the function $f(x) = |x-a| - |a|$, where in this case $a = v_2 - v_1$. We have, to the first order, $$f(x) \approx f(0) + \nabla f(0)\cdot x = \frac{a}{|a|}\cdot x,$$ hence $$|v_1 - v_2 - v_3| - |v_1 - v_2| = f(v_3) \approx \frac{-(v_1 - v_2)}{|v_2 - v_1|}\cdot v_3.$$