In solid states physics, there is an approximation that goes as follows: $$k_0|\vec{r}-\vec{R}|\approx k_0 \left(r-\frac{\vec{r}}{r} \cdot \vec{R}\right)$$ where $r$ is the norm of $\vec{r}$ and it is assumed that $r\gg R$.
I tried to start the derivation by writing out terms in $x,y,z$ but it became a mess and I was not clear how to proceed/start this derivation.
\begin{align} \vert \vec r-\vec R\vert&=\sqrt{r^2-2 \vec r\cdot \vec R + R^2}\, ,\\ &= r\sqrt{1-2\frac{\vec r\cdot \vec R}{r^2}+\frac{R^2}{r^2}}\, ,\\ &\approx r\left(1-2\frac{\vec r\cdot\vec R}{2r^2} \right) \end{align} using the binomial approximation $(1+x)^n\approx 1+n x$ when $\vert x\vert<< 1$. Note that $\vec r\cdot \vec R/r^2$ is of size $R/r$ whereas $R^2/r^2$ is obviously smaller when $r\gg R$.
