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

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

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 $(\vec{A} \times \vec{B}) \times \vec{C} = \dots$

$a \times (b \times c)$ is equal to

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