Equation of a Plane in R^n , n>3

[WWGD]

Hi,

Just curious: what is the equation of a plane in R^n for n>3 ?

We cannot define a normal vector for n>3 , so, what do we do? I thought

of working with flat, embedded copies of R^2 . In terms of linear algebra,

I guess we could see a plane in R^n as any image of a linear map with

rank=2. In terms of geometry, maybe we need a "flat" embedding of R^2

(I am not clear how to make the term 'flat' here more precise) in R^n.

Anyone know?

Thanks.

 

[Hootenanny]

Hi,

Just curious: what is the equation of a plane in R^n for n>3 ?

We cannot define a normal vector for n>3 , so, what do we do? I thought

of working with flat, embedded copies of R^2 . In terms of linear algebra,

I guess we could see a plane in R^n as any image of a linear map with

rank=2. In terms of geometry, maybe we need a "flat" embedding of R^2

(I am not clear how to make the term 'flat' here more precise) in R^n.

Anyone know?

Thanks.

In terms of $\mathbb{R}^n$, then for $\boldsymbol{x} = [x_1,x_2,\ldots,x_n]^\text{T}\in\mathbb{R}^n$ and non-zero scalars $a_n$ the sub-space

$$\text{const.} = \sum_{i=1}^n a_ix_i$$

is a hyperplane of $\mathbb{R}^n$. In actuality, the definition of a hyperplane is more compact: A hyperplane of any vector space is any vector subspace of co-dimension 1.

 

[WWGD]

Thganks, but I was thinking of a 2-d plane living in R^n with n higher than 2.

would that still be defined as a1.x1+a2.x2+a3.x3+0x4+...+0.xn=constant?

 

[HallsofIvy]

No. It cannot be done with a single equation like that. To identify an m-dimensional object in n-dimensional space requires n- m numerical equations. That is why Hootenanny was able to give a single equation for a hyper-plane (codimension 1 so dimension n- 1). To determine a 2 dimensional plane in n dimensional space would require n- 2 numerical equations.

 

[zhentil]

It would have n-2 linearly independent normal vectors. Take the intersection of the hyperplanes passing through a given point, each with one of the normal vectors.