If $\theta$ is the angle between the vectors $4 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and $\hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$, then $\sin 2 \theta=$

$\mathbf{a}, \mathbf{b}$ 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