Let $V, W$ be two affine subspaces of $\mathbb{R}^n$ with dimension $d$ and $n-d$, respectively. Suppose that they are not parallel, that is, when translated to the origin, it doesn't happen that $W \subset V$ or $V \subset W$.
Is it true, then, that $V$ and $W$ must intersect?
Thanks in advance!
Motivation: In Euclidean geometry and three dimensional plane geometry one can see that two affine subspaces of complementary dimension (that is of dimension d and n-d) respectively always intersect unless they are parallel in the sense that the translated vector subspaces are in containment relation. For example for $n=2$ this amounts to show that two non-parallel lines intersect or in $n=3$ case it becomes the question that a line and a plane not containing the line or any of its translates intersect nontrivially. It is natural to demand whether this has any analog in higher dimension also.
I don't think this is true. Consider $\mathbb{R}^4$ take the $XY$ plane translated at (0,0,0,1) and the $XZ$ plane. They are of complementary dimension but not intersecting. The idea was that two affine subspaces $a+V,b+W$ intersect if and only if $a-b \in V + W$. So I have chosen $V,W$ in a way so that their sum does not become $\mathbb{R}^n$. To meet your conditions I need to take $n=4$ as it seems that the result will be true for $n\leq 3$.