Consider two $2\times 1$ vectors, $v_1,v_2$.
As we know the convex combination of these two vectors, $r v_1+(1-r)v_2$, where $r\in [0,1]$ can be considered as a vector-valued function, $f: [0,1]\rightarrow \mathbb{R}^2$.
Also, on the $\mathbb{R}^2$, the function values lie between the two points $v_1$ and $v_2$ "linearly".
Here, why does the convex combination construct a "straight" line?