Why is the difference of two vectors parallel to the vector connecting the two vectors? Say we have any two vectors $u$ and $v$. We want to find a vector $w$ that is parallel to the vector that goes from $u$ to $v$ (is there a standard notation for this vector?). Then I already know that such a vector can be found simply by subtracting one of the two vectors from the other. Can you give me some intuition why this works? Probably what I am not understanding is how you get the vector connecting $u$ and $v$ in the first place.
 A: While there's already an accepted answer, I'd like to add some visual intuition to the ideas. In particular, I want to address the OP's sentence:

the vector that goes from $u$ to $v$


A very important aspect of vector spaces is that all vectors "start" from the origin. There is no such thing as a vector starting from the tip of another vector. What I have pictured in orange are not vectors, there are only helpful to get an intuition for vector addition. The "real" $u$, $v$ and $w$ vectors are in black, and sure, it's easy to understand $u+w=v$ if you follow the path (black $u$, orange $w$), but that doesn't mean that the orange $w$ is a member of the vector space. Same thing with $w=v-u$, which is $w=v+(-u)$.
A: It's a consequence of how vector addition works. Say we're traveling along $v$ then going to $u$ and to get there I travel along some other vector $x$. Geometrically we would model this as $v+x=u$ and then solving for $x$ gives us $x=u-v$. Now it becomes $v + (u-v)=u$ and we see this works both algebraically and geometrically.
