I'm self studying Feynman's Lectures on Physics, and he has a chapter devoted to vectors. At one point he discusses acceleration along a curve, and mentions that the change in velocity can be broken down into two components, one being tangential to the path.
It is interesting to note that we can compose the velocity difference out of two parts; we can think of acceleration as having two components, Δv∥, in the direction tangent to the path and Δv⊥ at right angles to the path, as indicated in Fig. 11–8. The acceleration tangent to the path is, of course, just the change in the length of the vector, i.e., the change in the speed v: a∥=dv/dt.(11.15)
The diagram shown (with colored additions by me) is: 
I expect the change in velocity (difference in length) to be the part of the arrow highlighted in blue, not the entire portion that is to the right of the vector $Δv_⊥$. What is wrong with my intuition?
(This can be moved to physics exchange if it's more appropriate there, but I felt that I'm just missing something about the mathematical model).