Answer
$arccos (-\frac{1}{3})\approx 109.5 ^\circ$
Work Step by Step
$a_1 \cdot a_2=|a_1||a_2| cos \theta$
$ \lt \frac{1}{2},-\frac{1}{2},-\frac{1}{2} \gt \cdot \lt -\frac{1}{2},\frac{1}{2},-\frac{1}{2} \gt=\sqrt {(\frac{1}{2})^2+(-\frac{1}{2})^2+(-\frac{1}{2})^2}\sqrt {(-\frac{1}{2})^2+(\frac{1}{2})^2+(-\frac{1}{2})^2} cos \theta$
$-\frac{1}{4}-\frac{1}{4}+\frac{1}{4}=\frac{3}{4} cos \theta$
$cos \theta =-\frac{1}{3}$
$ \theta = arccos (-\frac{1}{3})\approx 109.5 ^\circ$
