Use the dot product formula: ( mathbf{a} cdot mathbf{b} = |mathbf{a}|,|mathbf{b}| cos theta )
Solve for the angle: ( theta = cos^{-1}!left(dfrac{mathbf{a} cdot mathbf{b}}{|mathbf{a}|,|mathbf{b}|}right) )
Compute the dot product: ( mathbf{a} cdot mathbf{b} = a_1b_1 + a_2b_2 + cdots + a_nb_n )
Compute the magnitudes: ( |mathbf{a}| = sqrt{a_1^2 + a_2^2 + cdots + a_n^2} ), ( |mathbf{b}| = sqrt{b_1^2 + b_2^2 + cdots + b_n^2} )
Substitute the values into the angle formula
Ensure the result inside ( cos^{-1} ) is between (-1) and (1)
The angle is usually measured in radians unless converted to degrees
Convert to degrees if needed: ( theta^circ = theta times dfrac{180}{pi} )