The vectors $2 \hat{i}-3 \hat{j}+\hat{k}, \hat{i}-2 \hat{j}+3 \hat{k}$ and $3 \hat{i}+\hat{j}-2 \hat{k}$

The volume of a parallelepiped whose coterminous edges are $2 \overrightarrow{a}, 2 \overrightarrow{b}, 2 \overrightarrow{c}$ is:

If $\vec{u} = \hat{i} + 2\hat{j} - 2\hat{k}$, $\vec{v} = 2\hat{i} + \hat{k}$ and $\vec{w}$ is a unit vector, then the maximum value of the scalar triple product $[\vec{u} \vec{v} \vec{w}]$ is

The parallelepiped is determined by vectors $\vec{a} = -2\hat{i} + 5\hat{j} + 3\hat{k}$, $\vec{b} = \hat{i} + 3\hat{j} - 2\hat{k}$, and $\vec{c} = -3\hat{i} + \hat{j} + 4\hat{k}$. The altitude of the parallelepiped on the parallelogram base determined by vectors $\vec{b}$ and $\vec{c}$ is

The value of $\frac{[(\vec{a} \times \vec{b}) \times (\vec{b} \times \vec{c}), (\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a}), (\vec{c} \times \vec{a}) \times (\vec{a} \times \vec{b})]}{[\vec{a} \times \vec{b}, \vec{b} \times \vec{c}, \vec{c} \times \vec{a}]}$ is equal to

If the vectors $\overrightarrow{a}+\lambda \overrightarrow{b}+3 \overrightarrow{c}$,$-2 \overrightarrow{a}+3 \overrightarrow{b}-4 \overrightarrow{c}$ and $\overrightarrow{a}-3 \overrightarrow{b}+5 \overrightarrow{c}$ are coplanar,then the value of $\lambda$ is