Suppose $G$ is a semisimple $\mathbb{R}$-algebraic group with finite center, and suppose $G$ acts irreducibly on a vector space $V$. Suppose $U \subset V$ and $W \subset V$ are subspaces.

$\mathbf{Question 1}.$ Is it always possible to find $g \in G$ such that $g U$ and $W$ are in general position, i.e. $$\dim(g U \cap W) = \max(0, \dim U + \dim W - \dim V)?$$

I believe the answer to this question is no. I remember thinking about this when I was a graduate student, and there was some simple example, but I no longer remember what the example was. So the real question is: construct a simple counterexample to this assertion.

$\mathbf{Question 2}.$ It is possible to show using the Weyl unitary trick and orthogonality of characters that there is always $g \in G$ such that $$\dim(gU \cap W) \leq \frac{(\dim U)(\dim W)}{\dim V}.$$ (Actually John Stalker showed this to me when we were graduate students). The question is: can one get a better bound?

Basically, I am mostly curious if this circle of questions was ever studied, and what keywords do I need to look it up.