How to derive a general formula for unit vectors subtruction (which includes only the angles)

Question

Suppose I have two unit vectors: $v_1=(cos\phi_2,sin\phi_2),v_2 = (cos\phi_1,sin\phi_1)$

I want to find:

 1. The (small) angle between them
 2. The length of projection of $V_2$ on $V_1$

What is the mathematical expression for both calculations.

I know how to do this calculations by a series of actions, but, whenever I try to write a general mathematical formula (not an algorithm) involving only $\phi_1,\phi_2$ I get confused.

Any ideas?

Thank you!

Answer 1

The vectors are obtained rotating the unitary vector $\vec u=(1,0)$, the first, $\vec v_1$, by an amount of $\phi_2$ and the second, $\vec v_2$, of $\phi_1$. The angle between them is then $\phi_2-\phi_1$

The projection of $\vec v_2$ on $\vec v_1$ is, given $\vert\vec v_1\vert=\vert\vec v_2\vert=1$:

$$|\vec{v_2}|\cos\theta=\dfrac{\vec{v_1}.\vec{v_2}}{|\vec{v_1}|}=\vec{v_1}.\vec{v_2}=\cos\phi_2\cos\phi_2+\sin\phi_2\sin\phi_1=\cos(\phi_2-\phi_1)$$

As expected.