Matrix Multiplication not associative when matrices are vectors?

The issue is that, technically, $(a^T b)c$ doesn't exist either. You see, we often pretend $a^T b$ is the scalar $k:=a\cdot b$, but it's really a $1\times 1$ matrix whose only entry is $k$. It's one thing to left-multiply $c$ by $k$; it's another to left-multiply $c$ by the $1\times 1$ matrix itself, which you can't do. If each of these vectors has $n$ entries with $n\ne 1$, $(a\cdot b)c=kI_n c\ne kI_1 c$ ($I_1 c$ is of course undefined), where $I_m$ is the $m\times m$ identity matrix.