Let $\vec{a}=-\hat{i}-\hat{j}+\hat{k}$,$\vec{a} \cdot \vec{b}=1$ and $\vec{a} \times \vec{b}=\hat{i}-\hat{j}$. Then $\vec{a}-6 \vec{b}$ is equal to

If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors such that $\vec{a} \times (\vec{b} \times \vec{c}) = \frac{\vec{b} + \vec{c}}{\sqrt{2}}$,then the angle between $\vec{a}$ and $\vec{b}$ is:

If $\overline{a}, \overline{b}, \overline{c}$ are three vectors with magnitudes $\sqrt{3}, 1, 2$ respectively,such that $\overline{a} \times (\overline{a} \times \overline{c}) + 3 \overline{b} = \overline{0}$,and if $\theta$ is the angle between $\overline{a}$ and $\overline{c}$,then $\sec^2 \theta$ is:

Let $\vec{p}, \vec{q},$ and $\vec{r}$ be three non-coplanar unit vectors equally inclined to each other at an acute angle $\theta$. The value of $|\vec{p} \times (\vec{q} \times \vec{r})|$ is:

If $a=2 \hat{i}-3 \hat{j}+\hat{k}$, $b=\hat{i}-\hat{j}+2 \hat{k}$ and $c=2 \hat{i}+\hat{j}+\hat{k}$ are three vectors, then $|(a \times b) \times c|=$

If $a$ is a vector perpendicular to both $b$ and $c$,then