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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 $3D$.

ssane
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  • This is an equation describing a surface on a 3D coordinate system. The surface is the set of the points to satisfy the equation. while "living" in a 3D space, the plane is 2D. for example - $z=0$ is the x-y plane –  Jan 11 '20 at 17:01
  • Thanks for the comment. If we set $z=0$, it does not yield to a line in $2D$, rather than a plane in $2D$? – ssane Jan 11 '20 at 17:06
  • You have three coordinates in $ax_by+cz+d=0$, but one equation. So you effectively have two free parameters, say $x$ and $y$, which is what you expect for a 2D surface. – almagest Jan 11 '20 at 17:06
  • No, if you drop the $z$ parameter from the equation, then it can only describe planes parallel to the $z$-axis. In 3-D, a single linear equation always represents a plane and never a line. – amd Jan 11 '20 at 18:54

3 Answers3

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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 $ax+by+cz+d=0$).

Sam
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  • Thanks for the answer. I get your point. Line is one-dimensional, but each point in is it described with two parameters $x,y$, therefore each point is in two-dimensional space. Doesn't it conterintuitive? – ssane Jan 11 '20 at 17:34
  • @sane If this is counterintuitive, then you need to develop your intuitions about the meaning of “dimension” further. Within the line, you only need one number to identify a point, but this line lives on a plane, so to place any point correctly within that plane, whether or not it also belongs to some particular subset, you need two numbers. – amd Jan 11 '20 at 18:53
  • @ It is a matter of perspective really. If you could slightly change the way you visualize 3D geometry, it will become evident. – Sam Jan 12 '20 at 04:37
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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.

saulspatz
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  • Thanks. I see. Then, how we can describe a plane in $2D$? – ssane Jan 11 '20 at 19:00
  • @sane Well, in $2D$ the whole space is a plane, so each point of the plane is just given by its $(x,y)$ coordinates. It's just like a line in $1D$, really – saulspatz Jan 11 '20 at 19:06
  • Can't we have an explicit equation for this case? Okay, Probably, no! What is the equation for the line in 1D? (rhetorical question) – ssane Jan 11 '20 at 19:06
  • @sane Not really. The only reason for the equation in higher dimensions is that not all points lie on the plane, and we want to distinguish the ones that do. But if all points lie on the plane, there's nothing to distinguish. You could have an equation like $x+y=x+y$ (or $2+2=4$ for that matter,) that is satisfied no matter what $x$ and $y$ are, but I think you'll agree that this is pointless. – saulspatz Jan 11 '20 at 19:10
  • I see, Thanks!! – ssane Jan 11 '20 at 19:12
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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.