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?
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 $$Q=\{x_1=-1,\,x_2=0\}.$$