15. If $\vec{a}=\{\cos \alpha, \sin \alpha\}, \vec{b}=\{\cos \beta, \sin \beta\}$, and satisfy $|k \vec{a}+\vec{b}|=\sqrt{3} \mid \vec{a}-$ $k \vec{b} \mid(k>0)$.
(1) Express $\vec{a} \cdot \vec{b}$ in terms of $\alpha$ and $\beta$;
(2) Express $\vec{a} \cdot \vec{b}$ in terms of $k$;
(3) Find the minimum value of $\vec{a} \cdot \vec{b}$ and the angle $\theta$ between $\vec{a}$ and $\vec{b}$ at this time $(0 \leqslant \theta \leqslant \pi)$.
15. (1) $\vec{a} \cdot \vec{b}=\cos \alpha \cos \beta+\sin \alpha \sin \beta=\cos (\alpha-\beta)$;
(2) $|k \vec{a}+\vec{b}|^{2}=3|\vec{a}-k \vec{b}|^{2}$, i.e., $k^{2} \vec{a}^{2}+2 k \vec{a} \cdot \vec{b}+\vec{b}^{2}=3\left(\vec{a}^{2}-2 k \vec{a} \cdot \vec{b}+k^{2} \vec{b}^{2}\right)$, so
$$
\vec{a} \cdot \vec{b}=\frac{k^{2}+1}{4 k}(k>0)
$$
(3) $\vec{a} \cdot \vec{b}=\frac{k^{2}+1}{4 k} \geqslant \frac{1}{2}$, with equality holding if and only if $k=1$, at which point $\theta=60^{\circ}$.