If $\vec{a}=2 \hat{i}+\hat{j}+2 \hat{k},$ then the value of $|\hat{i} \times(\vec{a} \times \hat{i})|^{2}+|\hat{j} \times(\vec{a} \times \hat{j})|^{2}+|\hat{k} \times(\vec{a} \times \hat{k})|^{2}$ is equal to

If $\overline{a}=\frac{1}{\sqrt{10}}(3 \hat{i}+\hat{k})$ and $\overline{b}=\frac{1}{7}(2 \hat{i}+3 \hat{j}-6 \hat{k})$,then the value of $(\overline{a}-2 \overline{b}) \cdot \{(\overline{a} \times \overline{b}) \times (2 \overline{a}+\overline{b})\}$ is

Let $\vec{a}$ be a non-zero vector. If $\vec{x}=\hat{i} \times(\vec{a} \times \hat{i})$,$\vec{y}=\hat{j} \times(\vec{a} \times \hat{j})-\vec{a}$ and $\vec{z}=\hat{k} \times(\vec{a} \times \hat{k})-\vec{a}$,then $\left[\begin{array}{lll}\vec{x} & \vec{y} & \vec{z}\end{array}\right]=$

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

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 $\overline{a}=\frac{1}{\sqrt{10}}(4 \hat{i}-3 \hat{j}+\hat{k})$ 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