Solution
= Solution
The <dimension formula for a sum of subspaces> gives
$$
\dim(\mathcal U+\mathcal V)+\dim(\mathcal U\cap\mathcal V)=4.
$$
Since the sum lies in $\mathbb R^3$ and each plane has <dimension> two, the possibilities are
$$
\boxed{(\dim(\mathcal U+\mathcal V),\dim(\mathcal U\cap\mathcal V))=(2,2)\ \text{or}\ (3,1).}
$$
The first case means the planes coincide; two distinct planes through the origin intersect in a line.