I'm having some trouble setting up vector representations. I was given the following question:
Given are the points $A(1,-3)$, $B(4,2)$ and $C(-2,1)$. Set up the vector representation for the line $k$ through point $B$, which is parallel to $AC$.
After drawing the points, I had no idea where to start. I did the following:
$\overrightarrow{AC}=\bigg(\begin{array}{c} -3 \\ 4 \end{array}\bigg)$ and $\overrightarrow{CB}=\bigg(\begin{array}{c} 6 \\ 1 \end{array}\bigg)$. Let's say that $\overrightarrow{AC}=:\overline{q}$ and $\overrightarrow{CB}=:\overline{r}$. I first thought that the answer was:
$k: \bigg(\begin{array}{c} x \\ y \end{array}\bigg)=\overline{r}+\lambda\overline{q}$ with $\lambda\in\mathbb{R}$. Is this correct or did I make a mistake?
You made one mistake:
You can't use $\overrightarrow{CB}$ as a position vector on the line, since $C$ doesn't lie on $k$ (since $\overrightarrow{AC}\parallel k$).
A vector equation is
$$\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}4\\2\end{pmatrix}+\lambda\begin{pmatrix}-3\\4\end{pmatrix}$$
Since the line is parallel to $\overrightarrow{AC}$, we can use $\overrightarrow{AC}$ as a direction vector.
As a position vector, we can use $\overrightarrow{OB}=\begin{pmatrix}4\\2\end{pmatrix}$.