3. If $P$ is a point inside the cube $A B C D-E F G H$, and satisfies $P A=P B=\frac{3 \sqrt{3}}{2}$, $P F=P C=\frac{\sqrt{107}}{2}$, then the edge length of the cube is
3.5

Analysis: Let the side length of the cube be $a(a>0)$. Establish a coordinate system with $A$ as the origin, and $A B, A D, A E$ as the positive axes of $x, y, z$ respectively. Then $A(0,0,0), B(a, 0,0), C(a, a, 0), F(a, 0, a)$. Since $P A=P B$, $P$ lies on the perpendicular bisector plane of segment $A B$, which is $x=\frac{a}{2}$; since $P C=P F$, $P$ lies on the perpendicular bisector plane of segment $C F$, which is $y=z$, so we can set the coordinates of $P$ as $\left(\frac{a}{2}, y, y\right)$ $\left\{\begin{array}{l}\frac{a^{2}}{4}+y^{2}+y^{2}=\frac{27}{4} \\ \frac{a^{2}}{4}+(a-y)^{2}+y^{2}=\frac{107}{4}\end{array}\right.$, solving yields $\left\{\begin{array}{l}a=5 \\ y=\frac{1}{2}\end{array}\right.$ or $\left\{\begin{array}{l}a=\frac{4 \sqrt{6}}{3} \\ y=-\frac{7 \sqrt{6}}{12}\end{array}\right.$ (discard, since $y>0$)

Thus, $a=5$.