Question about $R^{2}$ and $R^{3}$

Question

I know that $R^{2}$ is a subset of $R^{3}$. If I have a 2-D cirle $A$ and also a 2-D rectangular polygon B that are both contained in $R^{3}$, would $ A \oplus B$ be 2-dimensional or 3-dimensional shape?

Answer 1

Since you are using the direct sum, I suppose that you think at $\mathbb{R}^2$ and $\mathbb{R}^3$ as vector spaces over $\mathbb{R}$. In this case,$\mathbb{R}^2$ is not a subspace of $\mathbb{R}^3$.

It is true that $\mathbb{R}^3$ contains infinitely many subsets( the ''planes'' in the ''space'') that are isomorphic to $\mathbb{R}^2$ (as a vector space), but the elements of $\mathbb{R}^3$, also if they are in a plane, are represented by three coordinates $(x,y,z)$.

For the last question: it is not clear what you means by the direct sum of a circle and a polygon but, intuitively, if the two figure stay in a same plane, the subspace that contains the two is this same plane, if they are in different planes then the subspace that contains the two is the whole $3$-D space.