In the general situation of $f:S\to R^m$ where $S\subset R^n$. There is a form of the mean value theorem: $a\cdot (f(y)-f(x))=a\cdot (f'(z)(y-x))$ which requires a vector $a$ and dot products.
Is it possible to create a generalization of the mean value theorem that doesn't involve dot products. Something like $$f(y)-f(x) = cf'(z)(y-x)$$ where $c$ is some real number. The idea being that the path $f(t(y-x)+ x)$ where $t\in [0,1]$ should have a tangent $f'(z)(y-x)$ that is parallel to $f(y)-f(x)$. Now there is no reason that they should have the same length though and so a constant scaling factor $c$ is needed. And to avoid some pathological cases maybe some conditions are needed such as $f(y)\neq f(x)$.