Let $a=2 \hat{i}+\hat{j}-3 \hat{k}$ and $b=\hat{i}+3 \hat{j}+2 \hat{k}$. Then the volume of the parallelopiped having coterminous edges as $a, b$ and $c$,where $c$ is the vector perpendicular to the plane of $a, b$ and $|c|=2$ 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:

If the vectors $a\hat{i}+\hat{j}+\hat{k}$,$\hat{i}+b\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+c\hat{k}$ are coplanar $(a \neq 1, b \neq 1, c \neq 1)$,then the value of $abc-(a+b+c)$ is:

If $\vec p$ and $\vec q$ are unit vectors such that $[\vec p, \vec q, \vec p \times \vec q] = \frac{1}{2}$,then the angle between $\vec p$ and $\vec q$ is

If the vectors $2i - 3j + 4k$, $i + 2j - k$ and $xi - j + 2k$ are coplanar, then $x = $

Let $\vec{b} = -\hat{i} + 4\hat{j} + 6\hat{k}$ and $\vec{c} = 2\hat{i} - 7\hat{j} - 10\hat{k}$. If $\vec{a}$ is a unit vector and the scalar triple product $[\vec{a} \ \vec{b} \ \vec{c}]$ has the greatest value,then $\vec{a}$ is equal to: