2E. Vector - Vector, Parametric equations Explanation (2D)

Linear equation

Proof - Find using $\vec{n}$ and 1 point

Let $\vec{n} = [\begin{matrix} a \\ b \end{matrix}]$ , $P = (x_{0}, y_{0})$ and $Q = (x, y)$ .

P and Q representation

P is a point we know, Q represents the point the equation solves

\begin{aligned} \vec{P Q} ⊥ \vec{n} & = 0 \\ \vec{P Q} \cdot \vec{n} & = 0 \\ [\begin{array}{c} x - x_{0} \\ y - y_{0} \end{array}] \cdot [\begin{array}{c} a \\ b \end{array}] & = 0 \\ a x - a x_{0} + b y - b y_{0} & = 0 \\ a x + b y & = a x_{0} + b y_{0} \end{aligned}