Consider the point $\bfx_0 = (x_0, y_0)$ and the line given by $\bfn \cdot \bfx = 0$, where $\bfn = (a, b)$. Using a vector projection, find the coordinates of the nearest point to $\bfx_0$ on the line $\bfn\cdot \bfx =0$.
In order to use a vector projection, we need to find a vector $\bfv$ such that the line $\bfn \cdot \bfx=0$ is given by all multiples of $\bfv$. The vector projection will then be the nearest point to $\bfx_0$:
We can take $\bfv$ to be any nonzero point that lies along the line. Hence we seek a single $\bfv$ such that $\bfn \cdot \bfv = 0$.