I'm reading a book about Vectors. I was in a problem where he showed two vectors ($\overrightarrow{AB}$ , $ \overrightarrow{AD}$ | $\overrightarrow{AB} \neq \vec{0} , \overrightarrow{AD} \neq \vec{0} ; \overrightarrow{AB} \nparallel \overrightarrow{AD})$.
And there is the relation: $ (1 - \alpha - \beta) \overrightarrow{AD} = \left( \frac{3}{4} \alpha - \frac{2}{5} \beta \right) \overrightarrow{AB} $
During the resolution of the problem, he wrote the following sentence:
As $\overrightarrow{AB}$ and $ \overrightarrow{AD}$ are not parallel, necessarily we should have: $$ 1 - \alpha - \beta = 0$$ $$ \frac{3}{4} \alpha - \frac{2}{5} \beta = 0$$
I think this happens, because this vectors aren't linearly dependent, so zero is the only number that satisfies the equation. But this concept wasn't written in the book. If it was there, what it would look like? I mean in a formal way.
Argue by contradiction. If $a\vec{u}=b\vec{v}$ and either $a$ or $b$ is not equal to zero, then either $\vec{u}=\frac{b}{a}\vec{v}$ or $\vec{v}=\frac{a}{b}\vec{u}$, respectively. So in either case, the vectors are parallel.