The OP should note that we usually denote a function from $V \times V$ to $V$ by a letter, say $f$, and write
$$ f: V \times V \to V$$
and denote the image of $(v,w) \in V \times V$ under $f$, an element in $V$, by $f(v,w)$.
Now if the operation is addition, $+$, of vectors, we find it both convenient and illuminating to use $\vec v + \vec w$ as opposed to $+(v,w)$.
Again, we might want to use some letter $f$ to denote a function from $\Bbb R \times V$ to $V$, so that if $\alpha \in \Bbb R$ and $v \in V$ the image under $f$ of $(\alpha,v)$ is denoted by $f(\alpha,v)$.
But if the operation is scalar multiplication of a vector, $\cdot$, we find it both convenient and illuminating to use $\alpha \cdot \vec v$ as opposed to
In fact, when multiplying by a scalar, the multiplication symbol is usually dropped when no ambiguity can occur, and juxtaposition of the scalar on the left of the vector is an appreciated notation/convention,
$$ \alpha \vec v$$
Note that in 1. Vector spaces of the book the author uses juxtaposition for scalar multiplication; see also the first example in 1.1. Examples.