Let $OA, OB, OC$ be the co-terminal edges of a rectangular parallelopiped of volume $V$ and let $P$ be the vertex opposite to $O$. Then, $[\overrightarrow{AP} \overrightarrow{BP} \overrightarrow{CP}]$ is equal to

$[b \times c, c \times a, a \times b]$ is equal to

Let $\vec{a} = \hat{i} - \hat{j}$,$\vec{b} = \hat{j} - \hat{k}$,and $\vec{c} = \hat{k} - \hat{i}$. If $\vec{d}$ is a unit vector such that $\vec{a} \cdot \vec{d} = 0 = [\vec{b} \, \vec{c} \, \vec{d}]$,then find $\vec{d}$.

Let $\vec{p}=2 \hat{i}+\hat{j}+3 \hat{k}$ and $\vec{q}=\hat{i}-\hat{j}+\hat{k}$. If for some real numbers $\alpha, \beta$ and $\gamma$,we have $15 \hat{i}+10 \hat{j}+6 \hat{k}=\alpha(2 \vec{p}+\vec{q})+\beta(\vec{p}-2 \vec{q})+\gamma(\vec{p} \times \vec{q})$,then the value of $\gamma$ is.

If $\overrightarrow{a}, \overrightarrow{b}$ and $\overrightarrow{c}$ are non-zero coplanar vectors,then $[2 \overrightarrow{a}-\overrightarrow{b} \quad 3 \overrightarrow{b}-\overrightarrow{c} \quad 4 \overrightarrow{c}-\overrightarrow{a}]$ is

If $\vec a = 3\vec j + 4\vec k$,$\vec b = 2\vec i + \vec k$ and $\vec c$,$\vec d$ are respectively the components of $\vec a$ parallel and perpendicular to $\vec b$,then the value of the scalar triple product $\left[ {(\vec a \times \vec c) \times (\vec c \times \vec d), (\vec c \times \vec d) \times (\vec d \times \vec a), (\vec d \times \vec a) \times (\vec a \times \vec c)} \right]$ is equal to: