Example 6 In a regular tetrahedron $ABCD$, $E$ and $F$ are on edges $AB$ and $AC$ respectively, satisfying $BE=3, EF=4$, and $EF$ is parallel to plane $BCD$. Then the area of $\triangle DEF$ is $\qquad$
(2017, National High School Mathematics Joint Competition (B Volume))
【Analysis】According to property (6), establish a rectangular coordinate system as shown in Figure 10.
Since the edge length of the regular tetrahedron is 7, we have
$A(0,0,0), B(7,0,0), C\left(\frac{7}{2}, \frac{7 \sqrt{3}}{2}, 0\right)$,
$D\left(\frac{7}{2}, \frac{7 \sqrt{3}}{6}, \frac{7 \sqrt{6}}{3}\right), E(4,0,0), F(2,2 \sqrt{3}, 0)$.
Since $\triangle D E F$ is an isosceles triangle, draw the altitude $D H$ from $D$ to the base $E F$, then $H$ is the midpoint of $E F$.
Thus, $H(3, \sqrt{3}, 0)$.
Using the distance formula between two points, we get $D H=\sqrt{33}$.
Therefore, $S_{\triangle D E F}=\frac{1}{2} E F \cdot D H=2 \sqrt{33}$.