You should have
[tex]L(t)=\begin{bmatrix}1\\1\end{bmatrix}+t\begin{bmatrix}2\\-3\end{bmatrix}[/tex]
[tex]\mathbf v[/tex] on its own points to the point (2, -3) and lies on the line through this point and the origin. We capture all points on this line by scaling [tex]\mathbf v[/tex] by [tex]t[/tex]. Then we add to [tex]t\mathbf v[/tex] another vector that points to P(1, 1), which shifts the endpoint of all vectors on the line [tex]t\mathbf v[/tex] to the line through P that runs parallel to [tex]t\mathbf v[/tex].
