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Example Find the angle between ${\bf A} = (1,1,2)$ and ${\bf B} = (-1,2,1)$.

Example

$\displaystyle \cos{\theta} = \frac{a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3}}{\Vert(... ...Vert(b_{1},b_{2},b_{3})\Vert} = \frac{-1+2+2}{\sqrt{6}\sqrt{6}} = \frac{1}{2}. $

Note that $\theta$ must be in th interval $\displaystyle{[0, \frac{\pi}{2}]}$. Thus, $\displaystyle{\theta = \frac{\pi}{3}}$.