Desired operation: A + B = C; {A,B,C) are vector quantities.
Issue: {A,B} contain error (measurement or otherwise)
Objective: estimate the error in the vector sum.
Let A = u + du; where u is the nominal value of A and du is the error in A
Let B = v + dv; where v is the nominal value of B and dv is the error in B
Let C = w + dw; where w is the nominal value of C and dw is the error in C [the objective]
C = A + B
w + dw = (u + du) + (v + dv)
w + dw = (u + v) + (du + dv)
w = u+v; dw = du + dv
The error associated with w is the vector sum of the errors associated with the measured quantities (u,v)