I am trying to understand how the following identity in homogeneous coordinates comes about and how to visualise the two multiplying parts $P$ and $l$ in the $3$D cartesian coordinates. Are they orthogonal to each other in order for the dot product to be zero. Can you show it in a picture?
Since any non-zero multiple of $l$ defines the same line, it is useful to consider $l$ as a vector of which only the direction matters. Let $p=(u,v,w)$ be a point in homogeneous coordinates. Then in order for $p$ to lie on the same line, the dot product of $p$ and $l$ must vanish, that is, $$p\cdot l = 0$$
My understanding is shown in the picture below.
More background can be found here:
We have seen that a point $(x,y)$ in the Cartesian plane has homogeneous coordinates $t(x,y,1)$, $t\neq0$. These coordinates would correspond to a line through the origin (excluded) if they were Cartesian coordinates in the $3$-dimensional space. When homogeneous coordinates are "viewed" as Cartesian coordinates, the dimensions of the geometric object they describe "increase" by one.
The geometry of a line in the Cartesian plane is reviewed in Appendix A. It has a general equation $ax+by+c=0$. Suppose $(u,v,w)$ are the homogeneous coordinates of a point $(x,y)$ on the line; hence $x=\frac{u}{w}$ and $y=\frac{v}{w}$. Substituting $x$ and $y$ in the line equation and multiplying through by $w$, yields the conditions for $(u,v,w)$ to be the homogeneous coordinates of a point on the line: $$au+bv+cw=0$$ The above equation is known as the homogeneous line equation. The line is uniquely specified by the coefficients $a,b$, and $c$, or any multiple $ra,rb$, and $rc$ with $r\neq0$. Therefore it is natural to specify the line by the homogeneous coordinates $$l=(a,b,c)$$