Context:
I am working through George Polya's book, Mathematics and Plausible Reasoning.
Let $P$ be a plane in $\mathbb{R}^3$ defined by $ax+by+cz=d$ where $a,b,c,$ and $d$ are constant real numbers.
My question:
What is the geometric significance and/or meaning of the constants $a,b,c,d$? What do they imply about the position and orientation of the plane in $\mathbb{R}^3$?
Attempted solution:
I see that, for a vector $(x,y,z)$ in the plane, it is required that $(a,b,c) \cdot (x,y,z) = |(a,b,c)||(x,y,z)|\cos(\theta)=d$. This seems to imply that varying $d$ and keeping $a,b,c$ constant might affect the "orientation," $\theta$, but then again $(x,y,z)$ can vary.
When $d=0$ and $(a,b,c),(x,y,z) \neq \vec{0}$, we have $\theta=\pi/2$, which implies that any vector in plane $P$ is always perpendicular to $(a,b,c)$.
On the other hand, if we consider a plane parallel to the $xy$ plane, $0x+0y+z=d,$ then it seems that $d$ determines how far along the z-axis the plane is.
At any rate, these are my scrambled thoughts so far. Could anyone help me see the bigger picture here?
When $d=0$, we have $ax+by+cz=(a,b,c)\cdot(x,y,z)=0$ which is satisfied by vectors $(x,y,z)$ orthogonal to the fixed vector $(a,b,c)$. Nonzero $d$ has the effect of shifting this plane in some direction. For example you could say it is shifted in the x direction by $ d/a$: $ax+by+cz=d \Leftrightarrow a(x-d/a)+by+cz=0$.