mean value theorem in several vars

My book says the MVT does not generalize to vector valued functions,
but the following version does:

a*[f(y) - f(x)] = a*[ f'(z)(y-x)]

where * means dot product and f'(z) is the total derivative of f at
z.

This looks just like the 1D version except for the dot product by a.
So what is a? It seems like for some fixed a, you get a 1D mean value
theorem in the direction of a. For example, if a = e1, the first unit
coordinate vector, then it is just a MVT in the first coordinate.

So for arbitrary a, is it just like doing a MVT application in each
coordinate and then taking a linear combination of them?

Geometrically, is this the same as the derivative at some point being
equal to the secant line, as in the 1D case, along the direction a?

.