A problem about affine spaces

Question

Let A be an affine space,dim(A)=4.

P,Q are planes from A.

If dir(P)!=dir(Q),then P and Q are disjoint.

Is this proposition true or false?

I know that two planes are parallel if they are disjoint only for an affine space B with dim(B)=2 or 3,but I don't know this assumption is true for 4-dimensional spaces.

Please,could you help me?

dir(P) is the vectorial space that lies below the 2-dimensional affine subspace P. — Adrian Costea, Mar 20 '16 at 20:11
And what would be $dim(P)$? Two or three? I it is two, then I'd try to adapt the case of two intersecting lines ${z=0,,x+y=0}$ and ${z=0,,x-y=0}$ in 3D-case to your 4D. — TZakrevskiy, Mar 20 '16 at 20:14

score 0 · Accepted Answer · answered Mar 20 '16 at 20:40

0

If $dir(P)=dir(Q)$, then $P$ and $Q$ can be either disjoint $P\cap Q=\emptyset$ or $P=Q$.

If $dir(P)\ne dir(Q)$, then we can find examples when $P\cap Q\ne \emptyset$ (take any planes $P$ and $Q$ such that $P=dir(P)$ and $Q=dir(Q)$) and when $P\cap Q=\emptyset$. Just take $$P=\{x_1=1,\,x_3=0\}$$ and $$Q=\{x_1=-1,\,x_2=0\}.$$

answered Mar 20 '16 at 20:40

TZakrevskiy

22,980

What is P={x1=1,x3=0}? – Adrian Costea Mar 20 '16 at 20:44
Please,could you tell me? – Adrian Costea Mar 20 '16 at 20:47

A problem about affine spaces

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