Let $a, b, c$ be vectors in $\mathbb{R}^3$. From what I understand, if $c\in \mathrm{Span}\{a,b\}$, then $b\in \mathrm{Span}\{a,c\}$.
Since they all fall on the same plane, I can't seem to find a counterexample
In what situation(s) would the statement be false?
If $c \in \operatorname{Span}(a,b)$, then there are constants $\alpha, \beta \in \mathbb{R}$ such that
$$c = \alpha a + \beta b$$
Rearrange this and find
$$b = \frac{1}{\beta} \left(c - \alpha a\right)$$
Now this isn't valid for $\beta = 0$, so it hints pretty strongly at possible counterexamples. (That is: What happens if $c = \alpha a$?)