Is taking sum inside cross product valid?

Question

I have a sum of a cross product over one of the multipliers. In this case it has a physics application being a sum over magnetic moments, $\vec{\mu}$, to give magnetisation, $\vec{M} = \sum_i\vec{\mu}_i$. What I'm doing is this $$ \sum_i \left(\vec{\mu}_i \times \vec{B}\right) = \vec{M} \times \vec{B}\,, $$ where $\vec{B}$ is the magnetic field and is taken to be independent of $i$. I just wanted to double check that taking the sum inside the cross product to replace the sum of $\vec{\mu_i}$ with $\vec{M}$ is valid?

Answer 1

The cross product is distributive, meaning that for any three vectors $\vec{x},\vec y,\vec z$, you have

$$(\vec x + \vec y)\times \vec z = (\vec x\times \vec z) + (\vec y\times\vec z)$$

This naturally extends to any finite number of vectors (you can prove this by induction), so if you have $\vec{x_1},\dots \vec{x_n}, \vec y$, you have

$$\sum_{i=1}^n(\vec{x_i}\times \vec y) = \left(\sum_{i=1}^n \vec{x_i}\right)\times \vec y$$