因為 $ABCD$ 與 $ABEF$ 均為正方形，且邊長為 $4$，我們有： $\overline{AB} = \overline{AD} = \overline{AF} = 4$ $\implies |\overset{\large\rightharpoonup}{AB}|^2 = 16$。 且 $\overset{\large\rightharpoonup}{AB} \cdot \overset{\large\rightharpoonup}{AD} = 0$， $\overset{\large\rightharpoonup}{AB} \cdot \overset{\large\rightharpoonup}{AF} = 0$。 根據幾何向量的合成： $\overset{\large\rightharpoonup}{AC} = \overset{\large\rightharpoonup}{AB} + \overset{\large\rightharpoonup}{AD}$ $\overset{\large\rightharpoonup}{AE} = \overset{\large\rightharpoonup}{AB} + \overset{\large\rightharpoonup}{AF}$ 計算內積 $\overset{\large\rightharpoonup}{AC} \cdot \overset{\large\rightharpoonup}{AE}$： $$\overset{\large\rightharpoonup}{AC} \cdot \overset{\large\rightharpoonup}{AE} = (\overset{\large\rightharpoonup}{AB} + \overset{\large\rightharpoonup}{AD}) \cdot (\overset{\large\rightharpoonup}{AB} + \overset{\large\rightharpoonup}{AF})$$ $$= \overset{\large\rightharpoonup}{AB} \cdot \overset{\large\rightharpoonup}{AB} + \overset{\large\rightharpoonup}{AB} \cdot \overset{\large\rightharpoonup}{AF} + \overset{\large\rightharpoonup}{AD} \cdot \overset{\large\rightharpoonup}{AB} + \overset{\large\rightharpoonup}{AD} \cdot \overset{\large\rightharpoonup}{AF}$$ $$= |\overset{\large\rightharpoonup}{AB}|^2 + 0 + 0 + \overset{\large\rightharpoonup}{AD} \cdot \overset{\large\rightharpoonup}{AF}$$ $$= 16 + 11 = 27$$ 故答案為 $27$。