The volume of a parallelepiped whose coterminous edges are represented by the vectors $\vec{OA}, \vec{OB},$ and $\vec{OC}$ with vertices $A(4, 3, 1), B(3, 1, 2),$ and $C(5, 2, 1)$ where $O$ is the origin,is ......... cubic units.

If $\bar{u}=\hat{\imath}-2 \hat{\jmath}+\hat{k}, \bar{v}=3 \hat{\imath}+\hat{k}$ and $\bar{w}=\hat{\jmath}-\hat{k}$,then the volume of the parallelepiped with $\bar{u} \times \bar{v}, \bar{u}+\bar{w}$ and $\bar{v}+\bar{w}$ as coterminus edges is

If $\overrightarrow x = 3\hat i - 6\hat j - \hat k$,$\overrightarrow y = \hat i + 4\hat j - 3\hat k$ and $\overrightarrow z = 3\hat i - 4\hat j - 12\hat k$,then the magnitude of the projection of $\overrightarrow x \times \overrightarrow y$ on $\overrightarrow z$ is

If $\bar{x}=\frac{\bar{b} \times \bar{c}}{[\bar{a} \bar{b} \bar{c}]}, \bar{y}=\frac{\bar{c} \times \bar{a}}{[\bar{a} \bar{b} \bar{c}]}$ and $\bar{z}=\frac{\bar{a} \times \bar{b}}{[\bar{a} \bar{b} \bar{c}]}$ where $\bar{a}, \bar{b}, \bar{c}$ are non-coplanar vectors,then the value of $\bar{x} \cdot(\bar{a}+\bar{b})+\bar{y} \cdot(\bar{b}+\bar{c})+\bar{z} \cdot(\bar{c}+\bar{a})$ is

If $|\vec{a}| = 4, |\vec{b}| = 3$ and $\vec{a} \cdot \vec{b} = 8$, then the scalar triple product $[\vec{a} \quad \vec{a} + \vec{b} \quad \vec{a} \times \vec{b}]$ is equal to:

If $a, b$ and $c$ are three non-coplanar vectors and $p, q$ and $r$ are vectors defined by $p=\frac{b \times c}{[a b c]}, q=\frac{c \times a}{[a b c]}, r=\frac{a \times b}{[a b c]}$,then $(a+b) \cdot p+(b+c) \cdot q+(c+a) \cdot r$ is