How can I plot a straight line which is in normal form? Take a look at this link.
Suppose, I have the following equation of a straight line:
$$x  \cos\theta+ y \sin \theta=p$$
If we rearrange the equation as follows:
$$y = \frac{p - x\cos\theta}{\sin\theta}$$
we are again stuck with infinite values of $x$. So, what is the benefit of introducing $\theta$ and $p$ here?
How can/should I plot this?
 A: Hint:
All you need to plot the line is to obtain two points on this  line.


*

*If $\sin\theta,\:\cos\theta \ne 0$, the $x$- and $y$-intercepts $A$ and $B$ are given by
$$x_A=\frac p{\cos\theta},\qquad y_B=\frac p{\sin\theta}.$$
Note: this equation is close tto the standard form of the equation of a straight line, given its $x$- and $y$-intercepts $a$ and $b$:
$$\frac xa+\frac yb=1.$$

*If $\cos\theta=0$, then $\sin\theta=1\:$ or $-1$, and the equation of the line is
$$y=p\quad\text{ or }\quad y=-p. $$

*If $\sin\theta=0$, then $\cos\theta=1\:$  and the equation of the line is $\;x=p$.

A: The normal form can be intepreted as follow, let


*

*$p$ the distance between the line and the origin

*$n=(\cos \theta, \sin \theta)$ the normal unitary vector to the line
therefore for any point $OP=(x,y)$ on the line by dot product we have
$$OP\cdot n=(x,y)\cdot (\cos \theta, \sin \theta)=x\cos \theta+y\sin \theta=p$$
Note that for a given line $\theta$ and $p$  are fixed therefore by the following
$$y = \frac{p - x\cos\theta}{\sin\theta}$$
for $\sin \theta \neq 0$ we can find all the points on the line.
A: A line’s normal is perpendicular to the line: if it makes an angle of $\theta$ with the $x$-axis, then the line itself makes an angle of $\theta+\pi/2$ with the $x$-axis. Now, $\cos(\theta+\pi/2)=\sin\theta$ and $\sin(\theta+\pi/2)=-\cos\theta$, so if your equation of the line has the form $ax+by+c=0$, then a direction vector of the line is $(b,-a)$. This, together with a known point on the line, which you can find by plugging in a convenient value for $x$ or $y$ and solving for the other variable, should be enough information for you to plot the line.
