How is the binomial expansion of the vectors? I'm trying to find out if there is an attempt to define binomial expansion of vectors. i.e
$$(\overrightarrow a + \overrightarrow b)^n = ?? $$
I tried to google around this (e.g : binomial expansion of vectors), but simple searches do not give any useful keywords as how this approach is named.
I'm trying to visualize how the following generalization goes. When two vectors are on the number line (one dimension), their sum is just the magnitude addition and the result is simply lying in the same number line. When the lines lie in general two dimensional space their 'sum' is equivalent to the square of the magnitude of their vector sum (which is essentially the binomial expansion of $(a + b)^2$ ). 
So how is the interpretation of $ (a+b)^3 $? Definitely it is not related to the 'normal' vector sum of the two vectors in 3d (that would be simply $(a + b)^2$).

You can take a binomial type expansion of the vector expression
$$|a-b|^n,$$
where $|\cdot|$ is the norm. For instance, in $\mathbb{R}^2$ consider $f(x) = |x+y|^n$ Taylor expanded about the point $x = (0, 0)$. For $n=2$ we have
\begin{align}
f(x) & = f(0) + x_1f_{x_1}(0) + x_2f_{x_2}(0) + \frac{1}{2!}\bigg(x_1^2f_{x_1x_1}(0) + 2x_1x_2f_{x_1x_2}(0) + x_2^2f_{x_2x_2}(0)\bigg) \\
& = |y|^2  + 2\langle x,y\rangle + |x|^2.
\end{align}
For $n = 4$ you would obtain
$$
|x+y|^4 = |y|^4 + 4 \langle x, y\rangle|y|^2 + (2|x|^2|y|^2 + 4\langle x, y \rangle^2) + 4 \langle x, y\rangle|x|^2 + |x|^4.
$$
