Plane Equation in R3

Question

I have watched many videos and read many articles on this subject, but I still don't seem to have any grasp of what this equation is actually telling us.

The equation of a plane in R3 is said to be:

Ax + By + Cz = D

But what does this equation actually tell us? What do 'Ax', 'By', 'Cz' and 'D' represent, and if we solve for 'Ax', 'By', 'Cz' and 'D' then what will we know?

A plane is stretches infinitely in the X,Y,Z directions so how will we understand the location of a plane just from this equation? Are we even trying to find the location of a plane? If not, then what is this equation?

What specifically is this equation going to tell us?

Answer 1

One of the ways to understand the equation is to plot it. Here for example is a Geogebra site where you can visualize a plane generated by an equation $A\cdot x+B\cdot y+C\cdot z=0$. https://www.geogebra.org/3d/dmvpja6f

As you can see (that is obviously not a proof) the equation does actually produce a plane. You can also notice that the numbers $A,B,C$ can be used to actually find a vector perpendicular to the plane. The vector is given by simply $(A,B,C)$. If you add the fourth term to the equation ("D"), it's effect is to move the plane in the direction of the vector described before. Without this term the plane will always hit the point $(0,0,0)$ however assuming $D \neq 0$ the point $(0,0,0)$ will certainly not lie in the plane. These are some of the main things you can get from such a description of the plane.

Answer 2

$A$, $B$, $D$ and $D$ are constants in this expression, meaning you know their value. With this is mind, the expression

$$ A x + B y + C z = D, \tag{1} $$

could be interpreted as the set of all points $(x, y, z)$ such that when you multiply $x$ by $A$, $y$ by $B$ and $z$ by $C$ and add these three numbers up, the result is exactly $D$.

As it turns out, when you try to put all the numbers $(x, y, z)$ satisfying this constraint in a cartesian axis, the result is a plane. This is an example: imagine $A=B=D = 0$ and $C =1$, in this scenario, the equation in (1) reduces to

$$ z = 0 $$

Again, we need to find all triplets of the form $(x, y, z)$ that satisfy $z = 0$, and that is exactly the set of points lying on the $xy$-plane