Find an explicit description of the line 3x+2y = 0 in R2 through the origin.
the answer is:
span {[2, -3]} , but how?
The line $\displaystyle 3x+2y=0$ in $\displaystyle \mathbb R^2$ is spanned by any nonzero vector $\displaystyle \begin{pmatrix}x \\ y\end{pmatrix}$ such that $\displaystyle (x,y)$ lies on the line. $\displaystyle \begin{pmatrix}2 \\ -3\end{pmatrix}$ is one such vector; another would be $\displaystyle \begin{pmatrix}-1 \\ \frac32\end{pmatrix}$.
I do not necessarily disagree with reply #2.
However, the point $\displaystyle (-2,3)$ is on the line $\displaystyle 3x+2y=0~.$
That line contains $\displaystyle (0,0)$, so its direction vector is $\displaystyle <-2,3>$.
Thus in traditional vector geometry its equation is $\displaystyle <0,0>+t<-2,3>~.$
So notation is everything here. Unless you tell us about the notation in use in your notes, we cannot really answer.