In the general situation of $f:S\to \mathbb R^m$ where $S\subset \mathbb R^n$. 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.
In Tom Apostol's Mathematical Analysis (Second Edition), page No. 355, I found that after choosing $a$ to be a unit vector and using Cauchy-Scwarz inequality, they have written $\parallel f(y)-f(x)\parallel\leq \parallel f'(z)\parallel \parallel y-x\parallel$. But how have they got rid of $a$ in the left hand side. If I choose the unit vector in the direction of the vector $f(y)-f(x)$, then it is possible, but how will it follow for an arbitrary unit vector? Please help!