Let's say $S$ is a $2$-dimensional vector space over the field of the real numbers with a weighted inner product
$$\langle x, y \rangle = x^\top Q y ,$$
where $Q$ is a positive-definite matrix. I want to "visualise" this space, so I can make sense of it.

When $Q$ is diagonal, $S$ is simply a version of $\mathbb{R}^2$ stretched along the Cartesian axes. This is easy to understand, so from now on assume $Q$ is not diagonal.

When $Q$ is not diagonal, it gets complicated, because two vectors that are perpendicular in $\mathbb{R}^2$, such as the standard basis vectors $e\_1$ and $e\_2$, are not perpendicular in the non-diagonal metric $Q$. In short, I have no idea how to make a mental picture of these vectors, or any other vector, in $S$.

One idea I have is to represent a vector $x$ in $\mathbb{R}^2$ along with a transformation $Tx$ which has the length and direction of $x$ in $S$, so I can see the difference between the two. Therefore, to find $T$, I think the first condition should be
$$
\begin{split}
\lVert Tx \rVert &= \sqrt{x^\top Q x} \\
\lVert Tx \rVert^2 &= x^\top Q x \\
(Tx)^\top Tx &= x^\top Q x \\
x^\top T^\top Tx &= x^\top Q x \implies T^\top T = Q
\end{split}
$$
But then I don't know how to continue. In particular, I don't know how to find the direction of $Tx$. Where does it point to? To me it seems that the problem is undetermined, and maybe the whole idea is misguided.