Direction Ratios of a Line in 2D

Question

Consider the general equation of a line in 2-D $ax+by+c=0$. How do I deduce the direction ratios of this line from the given format?

Answer 1

For $a,b\ne0,(-c/a,0),(0,-c/b)$ are two points lying on the line. So the line is parallel to their difference vector $(-c/a,c/b)$ i.e. the vector $(b,-a)$.

The same result is obtained by noting that the slope of the line $-a/b$ is equal to $y/x$ where $x,y$ are the d-ratios along $x,y$ axes respectively.

Answer 2

Here’s a neat way to think about it:

When $x$ is increased by, say, $1$, then $y$ must be decreased by $\frac ab$ to counter the change, as $$a(x+1) +b\left(y-\frac ab\right) +c =ax + a+by -a +c = ax+by+c=0$$ Hence, the direction vector of the line is $\left(1,- \frac ab\right)$ or equivalently $(b,-a)$.