Rate of change of vectors

Usually, when we talk in physics, all our 'basis' vectors are unit length. If we have a set of three orthonormal basis vectors of form ${ e_1, e_2 , e_3}$ then a vector in can be written as:

$$ V= sum_{i=0}^3 V^i e_i$$

Now suppose you were to differentiate this vector with time, let's say this was a velocity vector.

$$ frac{dV}{dt} = sum_{i=0}^3 frac{dV^i}{dt} e_i + V^i frac{de_i}{dt}$$

So, consider the $ frac{de_i}{dt}$ term, clearly since it's unit length by definition, it's length can not increase. This means that the only way that it can change with time is rotations.

These type of transformations are useful when you have rotating coordinate systems

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See this answer and linked post to question:

You are multiplying an infinitesimal by infinity and expecting the result to be meaningful when it is actually undefined. This is leading you astray.

There are various ways to see that the magnitude of the velocity is unchanged. My favourite is to write the magnitude as the dot product:

$$ |v|^2 = mathbf v cdot mathbf v $$

If we differentiate this wrt time we get:

$$ 2 |v| frac{d|v|}{dt} = 2 mathbf v cdot mathbf a $$

If $mathbf a$ is normal to $mathbf v$ then the dot product on the right hand side is zero. Since $|v| ne 0$ this means:

$$ frac{d|v|}{dt} = 0 $$