Problem 4. In triangle $A B C$ with the ratio of sides $A B: A C=4: 3$, the bisector of angle $B A C$ intersects side $B C$ at point $L$. Find the length of segment $A L$, if the length of the vector $3 \cdot \overrightarrow{A B}+4 \cdot \overrightarrow{A C}$ is 2016.
Answer: 288.

Solution. Note that by the property of the angle bisector of a triangle $B L: L C=B A: A C=4: 3$, from which $\overrightarrow{B L}=4 / 7 \cdot \overrightarrow{B C}$. Now

$$
\overrightarrow{A L}=\overrightarrow{A B}+\overrightarrow{B L}=\overrightarrow{A B}+\frac{4}{7} \cdot \overrightarrow{B C}=\overrightarrow{A B}+\frac{4}{7}(\overrightarrow{A C}-\overrightarrow{A B})=\frac{3}{7} \cdot \overrightarrow{A B}+\frac{4}{7} \cdot \overrightarrow{A C}=\frac{1}{7}(3 \cdot \overrightarrow{A B}+4 \cdot \overrightarrow{A C})
$$

Therefore, $|A L|=1 / 7 \cdot|3 \cdot \overrightarrow{A B}+4 \cdot \overrightarrow{A C}|=1 / 7 \cdot 2016=288$.