I know that for a linear code we have that the minimum distance $d$ under the Hamming metric is equal to the minimal weight of a non-zero element but this minimum weight is just the minimum distance to a particular element in the code (the zero element).
The proof of this just notices that if there is a minimal distance then it is the weight of the difference of two codewords $u''=u-u'$ which is itself a codeword as the code is linear.
I feel as if we ought to be able to generalise this but I've not been able to figure out how to do so.
Alright I figured this out, simply take the vector you want, lets call it $x$, then $d(x,x+u'')=w(x-(x+u''))=w(u'')=d(u,u')=d$.