Suppose that $V$ is a vector space over $\mathbb R$ (for simplicity) with addition denoted by $\oplus$ and scalar multiplication denoted by $\otimes$. Let $\mathbf u, \mathbf w \in V$ and let $\lambda \in \mathbb R$ and suppose we are asked to compute $$ \lambda \otimes \mathbf u \oplus \mathbf v $$ I was wondering how to do this. I would tend to say it is $$ (\lambda \otimes \mathbf u) \oplus \mathbf w $$ but it might as well be $$ \lambda \otimes (\mathbf u \oplus \mathbf w) $$ since, as far as I understand, none of the axioms for a vector space settles the precedence of scalar multiplication over vector addition. Is there a general rule for this ?
EDIT: I "know" that the cultural rule is precedence of scalar multiplication over vector addition. This is hinted at from shortcuts like $$ \lambda(\mathbf u + \mathbf v) = \lambda \mathbf u + \lambda \mathbf v $$ which (I think) should be written as $$ \lambda(\mathbf u+ \mathbf v) = (\lambda \mathbf u) + (\lambda \mathbf v) \,. $$ unless everyone agrees beforehand that scalar multiplication takes precedence. The thing I am wondering about is whether a computer (say) would be able to deduce this precedence from the axioms alone.
Thanks a lot for your help!