Example 1. As shown in Figure 1, the edge length of the cube is $\mathrm{a}$. (1) Find the angle $\theta$ between $\mathrm{AC}^{\prime}$ and $\mathbf{D B}^{\prime}$, (2) Draw a perpendicular line $\mathbf{A E}$ from $\mathbf{A}$ to the diagonal $\mathbf{A}^{\prime} \mathrm{C}$, prove that $\frac{\mathrm{AE}^{2}}{\mathrm{EC}^{2}}=\frac{1}{2}$.