$\mathbf{a}, \mathbf{b}, \mathbf{c}$ are unit vectors. If $\mathbf{a}, \mathbf{b}$ are perpendicular vectors, $(\mathbf{a}-\mathbf{c}) \cdot(\mathbf{b}+\mathbf{c})=0$ and $\mathbf{c}=l \mathbf{a}+m \mathbf{b}+n(\mathbf{a} \times \mathbf{b}) ;$ ( $l, m, n$ are scalars), then $n^2=$

If $O(0,0,0), A(1,2,1), B(2,1,3)$ and $C(-1,1,2)$ are the vertices of a tetrahedron, then the acute angle between its face $O A B$ and edge $B C$ is

If the angles between the sides of the $\triangle A B C$ formed by $A(2,3,5), B(-1,3,2)$ and $C(3,5,-2)$ are $\alpha, \beta$ and $\gamma$, then $\sin ^2 \alpha+\sin ^2 \beta+\sin ^2 \gamma=$

Let $2 \hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}}, 5 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}},-13 \hat{\mathbf{i}}-11 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$ be the position vectors of three points. $A, B$ and $C$, respectively. If $\mathbf{A B}=\lambda \mathbf{B C}$ and $\mathbf{A C}=\mu \mathbf{C B}$, then $\lambda+\mu=$