2D planes how it 2D inspite of it has 3 coordinates?

Question

We know that any point in the plane can be represented as an ordered pair $(a, b)$ of real numbers, where $a$ is the x-coordinate and $b$ is the y-coordinate.For this reason, a plane is called two-dimensional.

But why the equation of plane is $ax+by+cz+d=0$ which has 3 coordinates?

Answer 1

The dimension is not related to the total number of variables, but rather to the number of variables you can freely choose (degrees of freedom). For instance, if you pick some values for $x$ and $y$, you cannot freely choose the value of $z$: it will be determined by the equation.

For instance, the points that belong to the plane $x+y+z = 1$ can also be described as points of the form $$ (x,y,z) = (0,0,1) + t (1,0,-1) + s(0,1,-1), \quad t,s \in \mathbb{R} $$

So, choosing a point in that plane, amounts to deciding on the value of $t,s$, hence dimension 2.

Answer 2

The concept of a "plane" is intrinsic. Planes can exist inside other objects, and whatever is happening outside of that plane, i.e. whatever is extrinsic to that plane, is not particularly relevant to the fact that the plane is a plane.

So.... What is a plane?

Well, you can go back to axiomatic geometry, i.e. the axioms of Euclid as refined in the 19th century. Pasch's axiom says that if $P$ is a plane, and if $T$ is a triangle in $P$, and if $L$ is a line in $P$, and if $L$ does not contain a vertex of $T$, and if $L$ crosses one of the side of $T$, then $L$ must also cross one of the other two sides of $P$. I'm sure you can convince yourself that this is true in $$(*) \quad P = \{(x,y,x) \in \mathbb R^3 \mid ax + by + cz + d = 0\} \quad\text{(assuming $(a,b,c) \ne (0,0,0)$, of course)} $$

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For a more Cartesian point of view, it is true that the plane $P$ given by $(*)$ is a subset of the 3-dimensional Cartesian coordinate space $\mathbb R^3$. And it is true that each point of $P$ has three coordinates $(x,y,z)$. However, those three coordinates do not vary independently within $P$. What distinguishes a plane in $\mathbb R^3$ from the whole of $\mathbb R^3$ is that within $P$ one may find two coordinates which vary independently over $P$, but one cannot find three coordinates which vary independently over $P$. And this fact is true of any plane embedded in $\mathbb R^3$ that is given by a formula like $(*)$.

For example, if you consider the plane $P$ in $\mathbb R^3$ given by the equation $$3x + 2y + 5z = 7 $$ then the two coordinates $x,y$ vary independently over $P$, whereas the third coordinate $z$ is completely determined by the values of $x$ and $y$, which one sees easily by solving to get $z = -\frac{3}{5}x - \frac{2}{5} y + \frac{7}{5}$.