Inner product of two functions is constant means the magnitude of them both are constant and so is their angle

Question

I am unsure about how a constant inner product means that $\vert w(t)\vert,\vert v(t)\vert$ are of constant magnitude except if the parameterization of w and v choose vectors of the same magnitude. The definition of the vector field here is:

Answer 1

You have shown that $\langle v(t), w(t)\rangle$ is constant for all parallel vector fields $v(t)$ and $w(t)$. In the particular case, where we take $v(t)$ twice, this means that $\langle v(t), v(t)\rangle$ is constant. So we find that $v(t)$ has constant length along the curve. The reasoning for $w(t)$ is of course the same.