If $a = i + j - 2k$, then $\sum \{(a \times i) \times j\}^2$ is equal to

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 $\sin \theta$ is

If $\vec{A} = \hat{i} - 2\hat{j} - 3\hat{k}$,$\vec{B} = 2\hat{i} + \hat{j} - \hat{k}$,and $\vec{C} = \hat{i} + 3\hat{j} - 2\hat{k}$,then $(\vec{A} \times \vec{B}) \times \vec{C} = \dots$

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 :

If $\overline{a}=\frac{1}{\sqrt{10}}(4 \hat{i}-3 \hat{j}+\hat{k})$ and $\overline{b}=\frac{1}{3}(\hat{i}+2 \hat{j}+2 \hat{k})$,then the value of $(2 \bar{a}-\bar{b}) \cdot \{(\bar{a} \times \bar{b}) \times (\bar{a}+2 \bar{b})\}$ 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: