$u, v, n$ are vectors in $\mathbb{R}^{2}$, show that $(u-v)$ is $\perp n$

Question

Let $l : Ax+By+C= 0$ be a straight line. Let $\overrightarrow{u}=u_1\overrightarrow{i}+u_2\overrightarrow{j}$ and $\overrightarrow{v}=v_1\overrightarrow{i}+v_2\overrightarrow{j}$ be two points on $l$ with $\overrightarrow{u}-\overrightarrow{v}\neq 0$, $\overrightarrow{n}=A\overrightarrow{i}+B\overrightarrow{j}$. Show that$(\overrightarrow{u}-\overrightarrow{v}) \perp \overrightarrow{n}$.

Answer 1

Show that the scalar product $(\overrightarrow{u}-\overrightarrow{v}) \cdot \overrightarrow{n}=0$.

Use that $\overrightarrow{u} \cdot \overrightarrow{n}= -C$ and $\overrightarrow{v} \cdot \overrightarrow{n}= -C$.