In any $\triangle A B C, r^2 \cot \frac{A}{2} \cot \frac{B}{2} \cot \frac{C}{2}=$

Let the vectors $\mathbf{A B}=2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\mathbf{A C}=2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$ be two sides of a $\triangle A B C$. If $G$ is the centroid of $\triangle A B C$, then $\frac{27}{7}|\mathbf{A G}|^2+5=$

If $(\alpha, \beta, \gamma)$ is a triad of real numbers satisfying $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}=\alpha(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})+\beta(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})+\gamma(2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}})$, then $\alpha^2-\beta^2+\gamma^2=$

Let $L$ be a line passing through a point $A$ and parallel to the vector $2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$. Let $-7 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}+11 \hat{\mathbf{k}}$ be the position vector of a point $P$ on $L$ such that $|\mathbf{A P}|=12$. Then, the position vector of $\mathbf{A}$ can be