How would you determine how much error there is between a vector addition and the real results

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)