3. In the spatial rectangular coordinate system, the three vertices of $\triangle A B C$ have coordinates $A(3,4,1)$, $B(0,4,5)$, and $C(5,2,0)$. Then the value of $\tan \frac{A}{2}$ is ( ).
(A) $\sqrt{5}$
(B) $\frac{\sqrt{5}}{5}$
(C) $\sqrt{6}$
(D) $\frac{\sqrt{6}}{6}$
3. A.
$$
\begin{array}{l}
\text { Given } \overrightarrow{A B}=(-3,0,4), \overrightarrow{A C}=(2,-2,-1) \\
\Rightarrow \cos A=\frac{\overrightarrow{A B} \cdot \overrightarrow{A C}}{|\overrightarrow{A B}||\overrightarrow{A C}|}=\frac{-10}{5 \times 3}=-\frac{2}{3} \\
\Rightarrow \tan \frac{A}{2}=\sqrt{\frac{1-\cos A}{1+\cos A}}=\sqrt{5} .
\end{array}
$$