Q:

Conservation of energy in three dimension

I'm trying to derive the conservation of energy in 3D from the equation $\vec{F}=m\vec{a}$.
David Morin, in his book "Introduction to Classical Mechanics With Problems and Solutions" p. 138-139, proves the conservation of energy in 1D in the following way:

I wanted to prove the 3D version in the same way, so I got the term
$$\int_C m\vec{v} \cdot d\vec{v}$$
or, if parametrized,
$$\int_{t_0}^t m\vec{v}(t) \cdot \frac{d\vec{v}(t)}{dt} \ dt$$
This should obviously yield $$\frac{1}{2}m|\vec{v(t)}|^2 - \frac{1}{2}m|\vec{v(t_0)}|^2.$$
But what I'm wondering is, how do I deal with the dot product? Please see the diagram below.

The angle between $d\vec{v}$ and $\vec{v}$ looks too complicated to be taken into account at infinitesimal level. (and note that even $d\theta$ is not the angle between these two)

A:

You can write the $d\vec v$ in terms of the components along $\vec v$ and perpendicular to it.
$$d\vec v=d|\vec v| \hat v+v d\theta\hat\theta$$
When you multiply with $\vec v=|\vec v|\hat v$, the second term will wanish. So all you need to consider is the radial component (1 dimension)

