In a $\triangle A B C, A-B=120^{\circ}, R=8 r$, then $\frac{1+\cos C}{1-\cos C}=$

In $\triangle A B C, \sqrt{\frac{r \cdot r_2}{r_3 r_1}}=$

$\hat{\mathbf{i}}-2 \hat{\mathbf{j}}$ is a point on the line parallel to the vector $2 \hat{\mathbf{i}}+\hat{\mathbf{k}}$. If $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}$ is a point on the plane parallel to the vectors $2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\hat{\mathbf{i}}+2 \hat{\mathbf{k}}$, then the point of intersection of the line and the plane is

Points $P$ and $Q$ are given by $\mathbf{O P}=\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\mathbf{O Q}=-\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$. A line along the vector $\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}$ passes through the point $P$ and another line along the vector $\mathbf{b}=\hat{\mathbf{j}}-\hat{\mathbf{k}}$ passes through the point $Q$. If a line along the vector $\mathbf{c}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ intersects both the lines along the vectors $\mathbf{a}$ and $\mathbf{b}$ at $L$ and $M$, respectively, then $\mathbf{P M}=$