How do I find the angle between 2 vectors?

First, we need to recall 2 basic definitions of vector operations:

The dot product is defined on vectors u=[u₁, u₂,...u_n] and v=[v₁, v₂,..., v_n] as u . v = u₁v₁+u₂v₂+...+u_nv_n
The length (norm) of a vector v=[v₁, v₂,..., v_n] is the nonnegative scalar defined as ||v||=√(v . v)=√(v₁²+v₂²+...+v_n²)
Note that u & v must be the same size to compute the dot product.

Now the formula for the angle, θ, between 2 vectors is as follows:

            cos(θ)=(u . v)/(||u|| ||v||)

Notice that u & v can be any size so long as they are both the same size. That is, this formula can be used to find the angle between vectors in 2 dimensions and also to find the angle between vectors in 100 dimensions, however hard that is to imagine.

A handy rearrangement of that formula to isolate θ is:

θ=cos^-1( (u . v)/(||u|| ||v||) )