My text has an example as follows:
Three vectors, V U and W, in three dimension space.
They ask to locate two other vectors:
a) 1/3V + 1/3W + 1/3U
b) 1/2W + 1/2U
the answers:
a) falls in the middle of the triangle whose vertices are W, U, and V.
b) falls midway on the line between W and U
I stare at it and think, "yeah, ok, looks right" but honestly I cannot prove to myself why, and I also cannot think of other examples to look at to reinforce that I really understand this.
I guess the answer for 'a' is something like, "well, the vectors W and U and V are all points on a plane defined by the linear combinations of cW+dU+eV" Therefore the example c=d=e=1/3 is also going to be on that same plane. But I'm missing how that sticks it in the middle of that U-W-V triangle.
And similarly for the part-b question, it sounds right, but I don't have a relationship that I can exploit to prove it.
You can work the other way around. Take part (b), for example.
Given the vectors $W$ and $U$, points on the line joining $w$ with $u$ have position vectors $p = W + k(U-W)$, where k is a scalar.
This is because if you complete the vector addition triangle, $W+\overline{wu} = U$.
When k=0, $p=W$
When k=1, $p=W+U-W = U$.
So as you vary k from 0 to 1 you get points on the line $wu$ (you're doing a linear interpolation, really).
The midpoint is halfway along the vector from u to w (k=1/2), which shows you that the midpoint i.e. $p = W + (1/2) (U-W)$ = (1/2)W + (1/2) U, which is what part (b) asserts.