311. The lengths of vectors $\bar{a}$ and $\bar{b}$ are $a=8, b=5$ and the angle between the vectors $(\hat{a}, \bar{b})=150^{\circ}$. Find the scalar squares of the vectors and their scalar product.
Solution. Using formulas (8) and (10), we obtain the scalar product of the vectors

$$
\bar{a} \cdot \bar{b}=8 \cdot 5 \cos 150^{\circ}=40\left(-\frac{\sqrt{3}}{2}\right)=-20 \sqrt{3}
$$

the scalar squares of the vectors

$$
(\bar{a})^{2}=a^{2}=64,(\bar{b})^{2}=b^{2}=25
$$