Vector Equations of Lines 3

Find the distance of the point \(A(-3,1)\) to the line
\[\displaystyle{\vec{r}=\begin{bmatrix}1\\2\end{bmatrix}+\lambda \begin{bmatrix}-2\\3\end{bmatrix}}\]

using the method outlined below.

1. Find an expression for an arbitrary point \(D\) on the given line.
2. Using the expression you’ve produced, find a general expression for the vector \(\overrightarrow{AD}\).
3. Where \(\vec{b}\) is the direction vector of the given line, use the dot product \(\overrightarrow{AD}\cdot\vec{b}\) to find the coordinates of the point on the line closest to \(A\). Hence, find the distance from \(A\) to the line.