In a two-dimensional motion of a particle, the particle moves from point $A$, with position vector $\vec{r}_{1}$, to point $B$, with position vector $\vec{r}_{2}$. If the magnitudes of these vectors are, respectively, $r_{1}=3$ and $r_{2}=4$ and the angles they make with the $x$-axis are $\theta_{1}=75^{\circ}$ and $\theta_{2}=15^{\circ}$, respectively, then find the magnitude of the displacement vector.
(A) 15
(B) $\sqrt{13}$
(C) 17
(D) $\sqrt{15}$