Consider two vectors $x_1$ and $x_2$. Let $$a \equiv \frac{x_1 \cdot x_2}{||x_1||\,||x_2||}$$ denote a (transformation) of the angle between the two vectors, and let $$d\equiv ||x_1|| - ||x_2||$$ denote the difference in their lengths. Here $||\cdot||$ defines the Euclidean norm.
I am looking for a way to express the distance $$||x_1 - x_2||$$ in terms of $a$ and $d$ (if such a decomposition exists).
On
There cannot be such a formula because $a$ and $d$ do not uniquely determine $\|x_1-x_2\|$. This is because $a$ only describes the angle between $x_1$ and $x_2$. But there are infinitely many pairs of vectors with that angle and a difference in length $d$. You can easily see that taking a longer $x_1$ will also make $x_1-x_2$ longer.
No such expression is possible.
Consider, for example, two orthogonal vectors of length $r$. Both $a$ and $d$ are $0$, but they can not be used to determine the distance $r\sqrt{2}$ since it varies with $r$.