Taylor expansion of a vector function

[xBorisova]

Could someone please explain how does this taylor expansion work:
1/|r-r'| ≈ 1/r+(r.r')/r³

possibly you have to taylor expand twice to get this result, an attempt at which led me nowhere,
surely it cannot be this complicated.

any useful comment about this would be greatly appreciated

thank you

 

[TSny]

Hello, xBorisova. Welcome to PF!

As a hint, note that for any vector $\textbf{A}$, $|\textbf{A}| = \sqrt{\textbf{A}\cdot \textbf{A}}$.