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(B) Given $|\vec{p}| = 3$, $|\vec{q}| = 2$ and the angle $	heta = \pi/3$.$\vec{p} \cdot \vec{q} = |\vec{p}||\vec{q}| \cos(\pi/3) = 3 	imes 2 	imes

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$\sqrt{67} : \sqrt{63}$

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(B) Given $|\vec{p}| = 3$, $|\vec{q}| = 2$ and the angle $	heta = \pi/3$.$\vec{p} \cdot \vec{q} = |\vec{p}||\vec{q}| \cos(\pi/3) = 3 	imes 2 	imes (1/2) = 3$.Calculate $|\vec{a}|^2 = |3\vec{p} - \vec{q}|^2 = 9|\vec{p}|^2 + |\vec{q}|^2 - 6(\vec{p} \cdot \vec{q}) = 9(9) + 4 - 6(3) = 81 + 4 - 18 = 67$.So, $|\vec{a}| = \sqrt{67}$.Calculate $|\vec{b}|^2 = |\vec{p} + 3\vec{q}|^2 = |\vec{p}|^2 + 9|\vec{q}|^2 + 6(\vec{p} \cdot \vec{q}) = 9 + 9(4) + 6(3) = 9 + 36 + 18 = 63$.So, $|\vec{b}| = \sqrt{63}$.The ratio $|\vec{a}| : |\vec{b}| = \sqrt{67} : \sqrt{63}$.

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