Triangle

Geometry Level 4

See the picture above. It is known \(\overrightarrow{AB}=3\text{ cm}\), \(\overrightarrow{AC}=\dfrac{\sqrt{30}}{3}\text{ cm}\), \(\overrightarrow{AD}=4\text{ cm}\), and \(\overrightarrow{BC}=2\text{ cm}\). If \(\angle BAD=\alpha\), then \[ \sin\alpha=\frac{\sqrt{p}}{q}. \] Calculate \(\,p-q\).

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