# The plane has 3 variables and is 3 dimensional and the hyperplane has four variables

1. Dec 7, 2012

### s3a

In the following two problems I am trying to get a deeper intuition of, the plane has 3 variables and is 3 dimensional and the hyperplane has four variables and is 3 dimensional as well. Can someone please show me why, practically, in the context of these problems?

Question with hyperplane:
Find an equation of the hyperplane H in R^4 that passes through P(3, -4, 1, -2) and is normal to u = [2,5,-6,-3].

Question with (regular) plane:
Find an equation of the plane H in R^3 that contains P(1,-3,-4) and is parallel to the plane H' determined by the equation 3x - 6y + 5z = 2.

Any input would be greatly appreciated!

2. Dec 7, 2012

### Ray Vickson

Re: The plane has 3 variables and is 3 dimensional and the hyperplane has four variab

What is a hyperplane in R^n? (I know the answer, but I want you to spell it out.) What would be the general form of equation for a hyperplane in R^4? What do you get if you use the given data?

If you do not know what a hyperplane is, or what is its equation, you need to fill in that background knowledge first. Consulting a textbook or course notes (or even Google) would be step 1.

3. Dec 7, 2012

### s3a

Re: The plane has 3 variables and is 3 dimensional and the hyperplane has four variab

Hi, Ray.
A hyperplane, H, in R^n is the set of points (x_1, x_2, ..., x_n) that satisfy a linear equation

a_1 x_1 + a_2 x_2 + ... + a_n x_n = b

where the vector u = [a_1, a_2, ..., a_n] of coefficients is not zero (and is the normal vector).

4. Dec 7, 2012

### Michael Redei

Re: The plane has 3 variables and is 3 dimensional and the hyperplane has four variab

That's a good definition. Now, can you use Ray's suggestion and write down the general form of an equation for a hyperplane in R4? Can you get the coefficients a1,... from the data in your initial problem? How about the number b?

5. Dec 7, 2012

### s3a

Re: The plane has 3 variables and is 3 dimensional and the hyperplane has four variab

Sorry, I forgot to write that part!:

a_1 x_1 + a_2 x_2 + a_3 x_3 a_4 x_4 = b

I actually know how to do both problems; I'm just trying to understand what a hyperplane is and the (n – 1) dimensionality of it as well as what the differences between it and a (regular) plane are.

Question with hyperplane:
2x_1 + 5x_2 – 6x_3 – 3x_4 = –26

Question with (regular) plane:
3x – 6y + 5z = 1

6. Dec 7, 2012

### haruspex

Re: The plane has 3 variables and is 3 dimensional and the hyperplane has four variab

Not really. A plane in 3 space is essentially a 2 dimensional manifold, but interpreted as embedded in 3 dimensions. Once you take that view it all becomes consistent.

7. Dec 7, 2012

### s3a

Re: The plane has 3 variables and is 3 dimensional and the hyperplane has four variab

8. Dec 7, 2012

### Dick

Re: The plane has 3 variables and is 3 dimensional and the hyperplane has four variab

Can you elaborate on why you think "the plane has 3 variables and is 3 dimensional and the hyperplane has four variables and is 3 dimensional as well". You may be thinking of dimension differently in those two cases.

9. Dec 7, 2012

### Bipolarity

Re: The plane has 3 variables and is 3 dimensional and the hyperplane has four variab

Another way to thing about it is that a hyperplane of n-space is an (n-1)dimensional affine subspace of n-space.

So the hyperplane of a line is a point; the hyperplane of a plane is a line; the hyperplane of ambient 3-space is a plane, the hyperplane of 4-space would be an ambient 3-space... and so on. It becomes impossible to visualize in higher dimensions, so just use the lower dimensions to visualize.

A plane in 3-space has two free variables because it is constrained by exactly one equation in the 3 variables of the space.
In the same way, a hyperplane in n-space has (n-1) free variables because it is constrained by exactly one equation in the n variables of the space.
Thus, the hyperplane, having (n-1) free variables, is an (n-1) dimensional affine subspace of n-space.

BiP

10. Dec 8, 2012

### s3a

Re: The plane has 3 variables and is 3 dimensional and the hyperplane has four variab

I am very confused about this. I was just thinking that, for “normal” constructs, each additional variable implies an additional dimension and that the hyperplane has one less dimension than the amount of variables.[/QUOTE]

I don't understand what “affine” means. Okay so, when used in an English sentence, I must “take a hyperplane of something” rather than just having a hyperplane be a hyperplane like a plane is plane?

I don't see how a plane (in three dimensions) is constrained.

Did you just imply that a plane is also an (n – 1) dimensional construct (as opposed to just a hyperplane)?

Sorry for the dumb question(s) but, I am extremely lost with this all.

11. Dec 8, 2012

### Michael Redei

Re: The plane has 3 variables and is 3 dimensional and the hyperplane has four variab

A plane in 3-dimensional space consists of points with three coordinates (x,y,z). Since not each combination of these three is allowed, they're "restrained" by a linear equation, and such a plane is said to have the dimension "1 less than that of 3-dimensional space", i.e. dimension 2 (not 3).

It may help to remember that such a plane is just "a" plane, and not the one you think of as having just two coordinates (x,y) for each point. Each such plane in 3-dimensional space has all other properties of "the" plane, but they are not the same.

As to hyperplanes: they are just subsets of any n-dimensional space restricted by one linear equation. So in 4-dimensional space, a hyperplane has dimension 3, but in 17-dimensional space it would have dimension 16 etc.

And in 3-dimensional space, a hyperplane is any set of points satisfying one linear equation. This is, again, just "a" plane.

If you consider "the" plane with points (x,y), a linear equation describes a line, which is a hyperplane in the plane.

So, in a nutshell: "Hyperplane" doesn't mean "hyper = bigger (in dimension) than a plane", but "like a plane is to 3-dimensional space, but hyper = generalised to any number of dimensions".

12. Dec 8, 2012

### haruspex

Re: The plane has 3 variables and is 3 dimensional and the hyperplane has four variab

It means linear, but not necessarily through the origin.
If y = mx, we can say y is proportional to x, but if y = mx+c then it's an affine relationship.
Any relationship between co-ordinates constitutes a constraint. In 2D, the set of points satisfying the constraint x2+y2≤a2 is a disc. The set of points satisfying the constraint x2+y2=a2 is a circle. Mathematically, a circle in 2D is considered a one dimensional "manifold". If you are constrained to be on it, then at any given point on it there is only ever one dimension you can move in. We can only picture it in two or more dimensions, so it's natural to think of it as being 2 dimensional, but really that's because our ability to visualise is largely limited to embedding things in Euclidean space. In the same way, a 2-dimensional plane can be embedded in Euclidean 3-space, maybe orthogonal to an axis, maybe not, but it remains in essence a 2D object.

13. Dec 12, 2012

### s3a

Re: The plane has 3 variables and is 3 dimensional and the hyperplane has four variab

I believe that my confusion is more simple than all that. If the answer to my upcoming question below is “Yes.” then, I've overcome the trouble I've had when I initially started this thread.

Is it 100% correct to say that a hyperplane is just a term referring to any (n – 1) dimensional object in an n dimensional space whereas a (regular) plane is simply a hyperplane with the specific case of n = 3?

(Please provide a strict “Yes.” or “No.” prior to any potential elaborations.)

14. Dec 12, 2012

### Bipolarity

Re: The plane has 3 variables and is 3 dimensional and the hyperplane has four variab

Yes, with the added condition that the "object" must flat (or affine), i.e. linear.
If the object is not flat, it is not really considered a hyperplane.

For example, a sphere is a (3-1)dimensional object in 3 dimensional place. But it is not a hyperplane of 3-space, because it is not flat. This is obvious from the fact that the equation of a sphere is not linear.

But a plane in 3-space is indeed a hyperplane because it is flat, i.e. defined by a linear equation.

BiP

15. Dec 13, 2012

### Michael Redei

Re: The plane has 3 variables and is 3 dimensional and the hyperplane has four variab

That applies only to affine spaces, of course. In non-affine spaces the concept of "flatness" makes little sense.

16. Dec 13, 2012

### s3a

Re: The plane has 3 variables and is 3 dimensional and the hyperplane has four variab

Are hyperplanes with no adjective specified implicitly assumed to be affine? What I mean by this, if it's unclear, is that Wikipedia, for example, mentions affine hyperplanes, vector hyperplanes and projective hyperplanes, would a hyperplane automatically be assumed to be affine if an underlined part from above is not specified?

I would also like to understand/visualize how a(n) (affine) hyperplane separates the particular space into two half-spaces.

According to Wikipedia, it's above or below the constant term in the hyperplane equation but, I can't visualize that. (I'm referring to the two- and three-dimensional spaces since I expect to not be able to visualize four dimensions and more.)

17. Dec 13, 2012

### haruspex

Re: The plane has 3 variables and is 3 dimensional and the hyperplane has four variab

Projective hyperplanes are not Euclidean hyperplanes. They include a presentation of the 'points at infinity'. Certainly an unqualified hyperplane should be assumed to be affine.
An affine hyperplane is defined by v.x = a, where v is a given vector and the dot indicates dot product. It separates the space into the regions v.x < a and v.x > a. Any continuous path from a point x1 satisfying v.x1 < a to a point x2 satisfying v.x2 > a necessarily passes through some point x for which v.x = a

18. Dec 13, 2012

### Bipolarity

Re: The plane has 3 variables and is 3 dimensional and the hyperplane has four variab

I don't know about higher maths, but in linear algebra, my professor usually means "affine" hyperplane when he says hyperplane since most of my linear algebra deals with linear equations. In other branches of math, it is probably ambigious to just say hyperplane.

z = 0 represents the xy-plane, which is a hyperplane of $ℝ^{3}$. Above this plane is all the points in $ℝ^{3}$ for which z>0. On the other hand, below this plane is all the points in $ℝ^{3}$ for which z<0. In the equation z=0, the 0 is the constant term.

BiP

19. Dec 13, 2012

### Michael Redei

Re: The plane has 3 variables and is 3 dimensional and the hyperplane has four variab

That depends on the circumstances. When you're talking mainly about projective spaces, projective lines, projective planes etc., you'll usually omit the word "projective", since it's understood. And so a mere "hyperplane" would be taken to be a projective one.

20. Dec 13, 2012

### Michael Redei

Re: The plane has 3 variables and is 3 dimensional and the hyperplane has four variab

That only works if your scalar field is ordered, i.e. $\mathbb Q$ or $\mathbb R$.