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:

$A$ unit vector coplanar with $\hat{i}+\hat{j}+\hat{k}$ and $2\hat{i}+\hat{j}+\hat{k}$ and perpendicular to $\hat{i}+\hat{j}-\hat{k}$ is

Let $\overline{a}, \overline{b}$ and $\overline{c}$ be three non-zero vectors such that no two of them are collinear and $(\overline{a} \times \overline{b}) \times \overline{c} = \frac{1}{3}|\overline{b}||\overline{c}| \overline{a}$. If $\theta$ is the angle between vectors $\overline{b}$ and $\overline{c}$,then the value of $\operatorname{cosec} \theta$ is

Let $\vec{a}=-\hat{i}+\hat{j}+2\hat{k}$, $\vec{b}=\hat{i}-\hat{j}-3\hat{k}$, $\vec{c}=\vec{a}\times\vec{b}$ and $\vec{d}=\vec{c}\times\vec{a}$. Then $(\vec{a}-\vec{b}) \cdot \vec{d}$ is equal to :

For any vector $r$, the expression $i \times(r \times i) + j \times(r \times j) + k \times(r \times k)$ is equal to:

If $\vec{a}, \vec{b}, \vec{c}$ are three non-zero and non-coplanar vectors such that $\vec{a} \times (\vec{b} \times \vec{c}) = \frac{\vec{b}}{2}$, then the angle between $\vec{a}$ and $\vec{b}$ is ...