[tex]1.\ \overrightarrow{AB}\cdot\overrightarrow{AC}=3\times 5\times\cos\left(\dfrac\pi6\right)=15\times\dfrac{\sqrt 3}2=\dfrac{15\sqrt 3}2\;.[/tex]
[tex]2.\ \overrightarrow{AB}\cdot\overrightarrow{AC}=1\times 4\times\cos\left(\dfrac{8\pi}3\right)=4\times\cos\left(\dfrac{2\pi}3\right)=4\times\left(-\dfrac 12\right)=-2\;.[/tex]
[tex]3.\ \overrightarrow{AB}\cdot\overrightarrow{AC}=4\times 7\times\cos\left(\dfrac\pi4\right)=28\times\dfrac{\sqrt 2}2=14\sqrt 2\;.[/tex]
[tex]4.\ \overrightarrow{AB}\cdot\overrightarrow{AC}=2\times 2\times\cos\left(-\dfrac{5\pi}3\right)=4\times\cos\left(\dfrac \pi3\right)=4\times\dfrac 12=2\;.[/tex]
[tex]5.\ \overrightarrow{AB}\cdot\overrightarrow{AC}=7\times 10\times\cos\left(-\dfrac{7\pi}4\right)=70\times\cos\left(\dfrac \pi4\right)=70\times\dfrac{\sqrt 2}2=35\sqrt 2\;.[/tex]
