Why is vector parametrization of a line perpendicular to the line itself?

Question

For a vector of parameters $p$, a line can be expressed as $p^T + k = 0$. The vector p is perpendicular to that line. I can picture an example of that (for instance vector $(\frac{-1}2, 1)$ is perpendicular to the line it defines $y = \frac{1}2x$). However, I cannot understand why.

Answer 1

vector $ \vec{v} \in l \rightarrow \vec{v}\cdot \vec{p}=0 $ (dot product (scalar product) of vectors is equals 0)

For example $ [\vec{v}= (2,1) \in l :y = \frac{1}{2}x ] \rightarrow [(2,1)\cdot (\frac{-1}{2}, 1) = 2\cdot (-\frac{1}{2})+ 1\cdot 1 = -1 + 1 = 0] $