314. Find the angle $\varphi$ between two vectors

$$
\bar{a}=\bar{i}+\bar{j}-4 \bar{k} \text { and } \bar{b}=\bar{i}-2 \bar{j}+2 \bar{k} .
$$
Solution. We will use formula (16):

$$
\cos \varphi=\frac{\bar{a} \bar{b}}{a b}=\frac{1-2-8}{\sqrt{1+1+16} \sqrt{1+4+4}}=-\frac{\sqrt{2}}{2},
$$

therefore, $\varphi=135^{\circ}$.