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This is relatively simple because there is only one degree of freedom for 2D rotations. If v1 and v2 are normalised so that |v1|=|v2|=1, then, angle = acos(v1•v2)

where:
• = 'dot' product (see box on right of page).
acos = arc cos = inverse of cosine function see trigonometry page.
|v1|= magnitude of v1.

The only problem is, this won't give all possible values between 0° and 360°, or -180° and +180°. In other words, it won't tell us if v1 is ahead or behind v2, to go from v1 to v2 is the opposite direction from v2 to v1.