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## Homework Statement

I'm trying to show that ##\Sigma_j(\vec a_j \cdot \vec B)=\vec B\cdot\Sigma_j(\vec a_j)##

(I need this to be true to derive some angular momentum properties.)

**2. The attempt at a solution**

Let's say that in some coordinate system we can express the vectors as ##\vec B=<B_1,B_2,...>## and ##\vec a_j=<a_{j.1},a_{j.2},...>##

Then the sum of the dot products will be ##\Sigma_j(\vec a_j \cdot \vec B)=\Sigma_j(B_1a_{j.1})+\Sigma_j(B_2a_{j.2})+...=B_1\Sigma_j(a_{j.1})+B_2\Sigma_j(a_{j.2})+...=\vec B\cdot\Sigma_j(\vec a_j)##

I think the crux of my proof is that, since ##\Sigma_j(\vec B\cdot \vec a_j)## is a sum of a bunch of scalars, it should be independent of the coordinate system (because that's the definition of a scalar, right?).

I know this is a simple problem, but I have two questions:

First, is this a valid proof? (I've never proven a thing in my life )

Second, is there a way to show this without using a coordinate system?