If the points with position vectors $3i - 2j - k$,$2i + 3j - 4k$,$-i + j + 2k$,and $4i + 5j + \lambda k$ are coplanar,then $\lambda = \dots$

Let $\overline{a}, \overline{b}, \overline{c}$ be three non-coplanar vectors and $\overline{p}, \overline{q}, \overline{r}$ be defined by the relations $\overline{p}=\frac{\overline{b} \times \overline{c}}{[\overline{a} \overline{b} \overline{c}]}, \overline{q}=\frac{\overline{c} \times \overline{a}}{[\overline{a} \overline{b} \overline{c}]}, \overline{r}=\frac{\overline{a} \times \overline{b}}{[\overline{a} \overline{b} \overline{c}]}$. Then the value of the expression $(\overline{a}+\overline{b}) \cdot \overline{p}+(\overline{b}+\overline{c}) \cdot \overline{q}+(\overline{c}+\overline{a}) \cdot \overline{r}$ is equal to:

If the volume of the parallelepiped formed by three non-coplanar vectors $\vec{a}, \vec{b}$ and $\vec{c}$ is $4$ cubic units,then $[\vec{a} \times \vec{b} \quad \vec{b} \times \vec{c} \quad \vec{c} \times \vec{a}]$ is equal to

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non-zero non-coplanar vectors. Let the position vectors of four points $A, B, C$ and $D$ be $\vec{a}-\vec{b}+\vec{c}$,$\lambda \vec{a}-3 \vec{b}+4 \vec{c}$,$-\vec{a}+2 \vec{b}-3 \vec{c}$ and $2 \vec{a}-4 \vec{b}+6 \vec{c}$ respectively. If $\overrightarrow{AB}$,$\overrightarrow{AC}$ and $\overrightarrow{AD}$ are coplanar,then $\lambda$ is :

If three vectors $\vec{a} = 12\hat{i} + 4\hat{j} + 3\hat{k}$,$\vec{b} = 8\hat{i} - 12\hat{j} - 9\hat{k}$,and $\vec{c} = 33\hat{i} - 4\hat{j} - 24\hat{k}$ represent the coterminous edges of a parallelepiped,then its volume is:

Let $a = \hat{i} - 2\hat{j} + 3\hat{k}$ and $b = 2\hat{i} + \hat{j} + \hat{k}$. If $c$ is a unit vector such that $[a \ b \ c]$ is maximum, then $c =$