For which $p \in \Bbb R$ are vectors $\vec{a}=(p-1)\vec{i}+2\vec{j}$ and $\vec{b}=(p+4)\vec{i}+(p-2)\vec{j}$ collinear?
Well I know that $2$ vectors are $\textbf{non collinear}$ if and only if $\alpha\vec{a}+\beta\vec{b}=0 \iff \alpha=\beta=0$
So that would mean that if we wanted the linear representation of $2$ $\textbf{collinear}$ vectors to equal zero vector, either $\alpha \neq0$ or $\beta \neq 0$ or both of them are not zero.
$\alpha\vec{a}+\beta\vec{b}=0$
$1$) $\alpha\neq0$, $\beta =0$
$\Rightarrow \alpha\vec{a}+0\vec{b}=0$
This means $\vec{b}$ can be anything, and $\vec{a}=0$, am I right?
$\Rightarrow$ $(p-1)\vec{i}+2\vec{j}=0$
This has no solution?
$2$ ) $\alpha=0$, $\beta \neq0$
$\Rightarrow 0\vec{a}+\beta\vec{b}=0$
For this I also got no solution. And now I'm not sure what to do when $\alpha \neq0$, and $\beta \neq0$?