# Angle between two vectors

In there are two angles between any two vectors. The smallest one I'll call . The other one, its 360° complement, I'll call .

The smallest angle between two vectors and is given by: which follows easily from the geometric definition of the dot product: The other angle is simply the 360° complement: ## Oriented angle

Some applications call for the oriented angle . It's the angle going from one vector to the other in a specific direction. The convention that I use is that the direction be counter-clockwise.

Because the angle is oriented, changing the order of vectors (or the used convention) changes the angle's sign: or, equivalently: Furthermore: It's also worth noting that: which is why the dot product doesn't carry enough information to compute .

### In two dimensions

In any coordinate system: ### Derivation: in two dimensions

Using the geometric definitions of the two-dimensional cross product and the dot product: Solving for the sine and cosine: Solving for (see Systems of equations involving the cosine and the sine of an unknown): The simplification can be done because and eliminating it will not change the result (see properties of atan2). 