Rysunek pomocniczy:

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Mamy dane:

$\frac{\left|AD\right|}{\left|AB\right|}=\frac{2}{3}\Rightarrow \left|AD\right|=\frac{2}{3}\left|AB\right|\Rightarrow \left|AB\right|=\frac{3}{2}\left|AD\right|$  

$\frac{\left|AE\right|}{\left|AC\right|}=\frac{1}{5}\Rightarrow \left|AE\right|=\frac{1}{5}\left|AC\right|\Rightarrow \left|AC\right|=5\left|AE\right|$  

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Na podstawie rysunku widzimy, że:

$\overrightarrow{BC}=\overrightarrow{BA}+\overrightarrow{AC}=\overrightarrow{AC}+\overrightarrow{BA}=\overrightarrow{AC}-\overrightarrow{AB}=5\overrightarrow{AE}-\frac{3}{2}\overrightarrow{AD}$