$\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are three vectors such that $|a|=3,|b|=2 \sqrt{2},|c|=5$ and $\mathbf{c}$ is perpendicular to the plane of $\mathbf{a}$ and $\mathbf{b}$. If the angle between the vectors a and $\mathbf{b}$ is $\frac{\pi}{4}$, then $|\mathbf{a}+\mathbf{b}+\mathbf{c}|=$

If $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ are non-coplanar vectors and the points $\lambda \mathbf{a}+3 \mathbf{b}-\mathbf{c}, \mathbf{a}-\lambda \mathbf{b}+3 \mathbf{c}, 3 \mathbf{a}+4 \mathbf{b}-\lambda \mathbf{c}$ and $\mathbf{a}-6 b+6 \mathbf{c}$ are coplanar, then one of the values of $\lambda$ is

If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are unit vectors such that $\mathbf{a}$ is perpendicular to both $\mathbf{b}, \mathbf{c}$ and angle between $\mathbf{b}, \mathbf{c}$ is $2 \pi / 3$, then $|a+3 b-4 c|^2=$

Let $\mathbf{a}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ be the position vector of a point $A$. Let $\mathbf{b}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\mathbf{c}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ be two vectors and $\mathbf{r}$ be a vector passing through the point $A(\mathbf{a})$ and parallel to the vector $\mathbf{b}$. If the projection of $\mathbf{r}$ on $\mathbf{c}$ is $\frac{9}{\sqrt{6}}$, then $|\mathbf{r}|=$