Understanding the equation of plane

Question

I have a simple question regarding plane (sorry, if it may sound incorrect, I am confused to understand it).

In general, plane is a two-dimensional surface that extends infinitely far. If it is two-dimensional surface, then each point in the surface can be described by two parameters, say $x$ and $y$. From another hand, the general equation of plane is given by $$ax+by+cz+d=0.$$ My question is: if plane is a two-dimensional surface, why we need the third $z$ parameter to describe it? Can someone give me a clear intuition of the equation?

P.S. I do recognize the if we drop the $z$ parameter from the equation we will end up with a line in $2D$, however I find a confusion that plane is two-dimensional surface, but it is needed to describe it in $3D$.

Answer 1

You are thinking about this wrong. If we consider a plane in $3$-dimensional space, each point of the space has three coordinates: $(x,y,z)$. To specify a plane, we have to say what is the relation between $x,y,\text{ and }z$, so we have three variables. (That is to say, given a point $(x,yz)$ in $3$-space, we want to be able to say if it lies on the plane by inspecting the coordinates. We need all three of them.) The equation of a line in two-space has two variables, fo the same reason. If we look at a line in one-space, it's just the real line, and we need only one coordinate to specify a point.

I guess you haven't learned about parametric equations yet. When you do, you'll see that it's possible to describe a plane in three-space with only two variables, but the idea is a little different.

Answer 2

Following your logic, a line being one-dimensional should only require a single parameter to describe it. But this is incorrect as you need two parameters.

For a line, you need a measure for the slope ($-\frac ab$) and another for positioning that line within the plane (the $c$ part in $ax+by+c=0$).

Similarly for a plane, you need two parameters for the 3D slope and another for the position of the plane in space (the $d$ part in $ax+by+cz+d=0$).

Answer 3

For me, the intuition is that there are $3$ variables, $x,y$ and $z$, but instead of $3$ degrees of freedom, the equation cuts it down to $2$. Two degrees of freedom is what we have in a plane.

Again, choose values for two of the variables freely, and the third variable is determined.