If $A, B, C, D$ are points in the plane $R^3$, let $\overrightarrow{u}=\overrightarrow{AB}$ and $\overrightarrow{v}= \overrightarrow{CD}$ (Note: indeed, we can consider any euclidean space, but I am setting the $R^3$ to get intuitive ideas better). 

Does it then make sense to add $\overrightarrow{u}+\overrightarrow{v}$? 

For me, it doesn't make sense since to add two vectors they should start from the same point. On one hand, from the point of view of Manifold Theory (which I guess is the real theory behind this), the reason why we cannot add them is because they belong to two different *tangent spaces* (actually two different *vector spaces*), where the tangent spaces are based on $A$ and $C$ respectively. 

On the other hand, I have heard some people say that we are able to add them because we can move one of the vectors, say $\overrightarrow{v}$, in such a way to make his starting point coincide with $A$ (the starting point of $\overrightarrow{u}$). This new vector, call it $\overrightarrow{v\;'}$, is just a translation of $v$ and thus they are thought to be equivalent and because of this we can do the addition. Another way that some people justify the addition of $\overrightarrow{u}$ and $\overrightarrow{v}$ is because (they say) $\overrightarrow{u}=B-A$ and $\overrightarrow{v}=D-C$ and then we are treating them as vectors starting from the origin and of course having vectors like those we can add them. 

What I think about the meaning of $\overrightarrow{u}=B-A$ (and respectively for $\overrightarrow{v}=D-C$) is because $\overrightarrow{u}$ has coordinates with respect to the canonical basis in the tangent space $T\_A R^3$ given by $B-A$. Also, the coordinates of $\overrightarrow{v\;'}$ in the tangent space $T\_A R^3$ are $D-C$ and these is the reason why we consider $\overrightarrow{v}$ and $\overrightarrow{v\;'}$ equivalent. 

Where does this equivalence make sense? I think just for geometrical purposes. For instance, when we talk about a normal vector to some plane (contained in $R^3$), we represent this by drawing a vector with its starting point belonging to the given plane. When we draw it we set the normal vector starting at ANY point of the plane, but if the plane has equation $ax+by+cz+d=0$, we say that the normal vector is $(a,b,c)$, which is a vector with its starting being the origin! 

What I think here that makes sense to all of these is because when we draw the normal vector starting at any point of the plane, what we mean is that no matter what vector we can pick with the property of being perpendicular to the plane, then its coordinates in the tangent space of its starting point are exactly $(a,b,c)$.

Now, I have another way to justify why it only makes sense when the vectors start at the same point. From Physics, if we have two forces acting on some object, we can add them to get the total force, but this makes sense because we have the two forces acting at the some point, however, if we have one force ($\overrightarrow{u}$) acting on one point ($A$) and another force ($\overrightarrow{v}$) acting on another point ($C)$, what sense does it have to add them? No one, I guess.

I hope my concern is clear and the reasons explained above also.

Thanks in advance.