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

If $\vec{a}$ and $\vec{b}$ are vectors in space given by $\vec{a}=\frac{\hat{i}-2 \hat{j}}{\sqrt{5}}$ and $\vec{b}=\frac{2 \hat{i}+\hat{j}+3 \hat{k}}{\sqrt{14}}$,then the value of $(2 \vec{a}+\vec{b}) \cdot[(\vec{a} \times \vec{b}) \times(\vec{a}-2 \vec{b})]$ is

Let $\bar{a}=\hat{i}+\hat{j}+\hat{k}$,$\bar{b}$ and $\bar{c}=\hat{j}-\hat{k}$ be three vectors such that $\bar{a} \times \bar{b}=\bar{c}$ and $\bar{a} \cdot \bar{c}=0$. If the length of the projection vector of the vector $\bar{b}$ on the vector $\bar{a} \times \bar{c}$ is $l$,then the value of $3l^2$ is

Let $\overline{a}, \overline{b},$ and $\overline{c}$ be non-zero vectors such that $(\overline{a} \times \overline{b}) \times \overline{c} = \frac{1}{3} |\overline{b}| |\overline{c}| \overline{a}.$ If $\theta$ is the angle between $\overline{b}$ and $\overline{c},$ then $\sin \theta = .....$

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 $(\bar{a} \times \bar{b}) \times \bar{c} = -5 \bar{a} + 4 \bar{b}$ and $\bar{a} \cdot \bar{b} = 3$,then the value of $\bar{a} \times (\bar{b} \times \bar{c})$ is