The Average of a vector square minus the square of the vector averaged Say I've obtained a series of 3D vectors, (x, y, z), with a different vector at different points in time. I want to obtain the second moment of this set of vector's fluctuations in time. This is given by: 
$<$ M $^2> - <$ M $>^2$,
where M is the vector in 3D, and the average is over time. To me, you can solve this simply by taking:
$<$ M $^2> - <$ M $>^2 = <$ (x, y, z )  $ \cdot $ (x, y, z ) $> - <$ (x, y, z ) $> \cdot <$ (x, y, z ) $>$.
However, I have seen this solved by decomposing the vector along different Cartesian axis:
$<$ M $^2> - <$ M $>^2 = < $ M $_x^2 > - < $ M $_x >^2 + < $ M $_y^2 > - < $ M $_y >^2 + < $ M $_z^2 > - < $ M $_z >^2 $.
This is shown on page 2548 in Pitera et al., Biophysical Journal, 2001, 20, 2546–2555. Paper Link.
I just wanted some clarification if this second method is correct, and I guess whether the first is also. To me, the second method is not possible as you cannot decompose $<$ M $>$ as they have. Less so if you then square the decomposed parts.
Thank you for any help, and apologies for any grievances about the formatting.

Okay let's see. Note that $M=M_xe_x+M_ye_y+M_ze_z \in \mathcal{R}^3$ (note that I write $M=(M_x,M_y,M_z)^T$ since $(x,y,z)^T$ is just the position vector in $\mathcal{R}^3$ and you are looking for the vector $M$, so look at the components of this vector). Calculating the variance of $M$ gives:
$\sigma_M^2=\langle M^2\rangle-\langle M\rangle^2=\langle M_x^2+M_y^2+M_z^2\rangle-\langle M_xe_x+M_ye_y+M_ze_z\rangle \cdot \langle M_xe_x+M_ye_y+M_ze_z\rangle = \langle M_x^2\rangle+\langle M_y^2\rangle+\langle M_z^2\rangle - \langle M_x\rangle^2 - \langle M_y\rangle^2 -\langle M_z\rangle^2$ where we used that $\langle M_xe_x + M_ye_y +M_ze_z\rangle=\langle M_x\rangle e_x + \langle M_y\rangle e_y + \langle M_z\rangle e_z$ (by the rules of expectation values, you can split it out and vectors have constant coefficients) and just worked out the (standard) inner product on $\mathcal{R}^3$ between two of such vectors. Hope it helps, it is a basic question on variances in principle. Greets.
