1. In the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, the size of the dihedral angle $B-A_{1} C-D$ is $\qquad$
(2016, National High School Mathematics League Fujian Province Preliminary)
Hint Establish a spatial rectangular coordinate system as shown in Figure 11.
Let the edge length of the cube be 1. Then
$$
\begin{array}{l}
D(0,0,0), A(0,1,0), B(1,1,0), \\
C(1,0,0), A_{1}(0,1,1) .
\end{array}
$$

By property (7), the equation of plane $B A_{1} C$ is
$$
x+z-1=0 \text {, }
$$

and the equation of plane $A_{1} C D$ is $y-z=0$.
The cosine of the angle between the normal vectors of the two planes is
$$
\cos \theta=-\frac{1}{2} \text {. }
$$

Therefore, the dihedral angle $B-A_{1} C-D$ is $120^{\circ}$.