I have a question about vector notation. In physics class today, my professor introduced the concept of average velocity. He defined it mathematically as
$$\vec{v} := \frac{\Delta \vec{x}}{\Delta \textrm{t}},$$
where $\Delta \vec{x}$ indicates the change in displacement over a time interval $\Delta \textrm{t}$.
However, in my previous math classes, I was always told that the division of a vector by a scalar was not a defined operation. For example, if I wanted to divide a vector $\vec{a}$ by a certain scalar $b$, I would write it as $\frac{1}{b}*\vec{a}$, not $\frac{\vec{a}}{b}$. Applying this to average velocity would entail that its definition should be $$\vec{v} := \frac{1}{\Delta \textrm{t}}*\Delta \vec{x}.$$
Is this a valid statement? Or am I misunderstanding the vector/scalar division comment from my previous classes?