How to approach finding lines that are perpendicular to each other with as little calculations as possible?

Question

Say we are given equation of a line in $ \mathbb{R^2}$ : $$k:2x+4y-1=0$$Now we want to find a line that is perpendicular to the line $k$ with as little calculations and transformations as possible, so that we can find it relatively quickly,

How one would go about doing that? - I'd suggest using vectors, but I, myself have trouble with expressing these relations simply...

Answer 1

Note that the vector of the line $k$ is $n=(2,-1)$. Assume that the perpendicular line passes the point $(x_1,y_1)$, then, in vector form, its equation is given by

$$(x-x_1,y-y_1)\cdot n=0$$

Answer 2

If your question is limited to $\mathbb{R}^2$ you can

1. find the slope $m$
2. take all lines with slope $-1/m$ of the form $y=b -x/m$ for any $b$.