Let $\mathbf{a}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and $\mathbf{b}=\hat{\mathbf{i}}+\hat{\mathbf{j}}$ be two vectors. If $\mathbf{c}^{\text {is }}$ vector such that $\mathbf{a} \cdot \mathbf{c}=|\mathbf{c}|,|\mathbf{c}-\mathbf{a}|=2 \sqrt{2}$ and the angle between $\mathbf{a} \times \mathbf{b}$ and $\mathbf{c}$ is $30^{\circ}$, then $|(\mathbf{a} \times \mathbf{b}) \times \mathbf{c}|=$

$$ (\mathbf{a}+2 \mathbf{b}-\mathbf{c}) \cdot(\mathbf{a}-\mathbf{b}) \times(\mathbf{a}-\mathbf{b}-\mathbf{c})= $$

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 $\mathbf{P M}=$

For $a \in R$, if the vectors $\mathbf{p}=(a+1) \hat{\mathbf{i}}+a \hat{\mathbf{j}}+a \hat{\mathbf{k}}$, $\mathbf{q}=a \hat{\mathbf{i}}+(a+1) \hat{\mathbf{j}}+a \hat{\mathbf{k}}$ and $\mathbf{r}=a \hat{\mathbf{i}}+a \hat{\mathbf{j}}+(a+1) \hat{\mathbf{k}}$ are coplanar and $3(\mathbf{p} \cdot \mathbf{q})^2-\lambda|\mathbf{r} \times \mathbf{q}|^2=0$, then the value of $\lambda$ is