If $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$ and $\vec{b} = \hat{i} \times (\vec{a} \times \hat{i}) + \hat{j} \times (\vec{a} \times \hat{j}) + \hat{k} \times (\vec{a} \times \hat{k})$,then $|\vec{b}|$ is

Which of the following statements is true regarding the vector triple product $(a \times b) \times c$?

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

$\overrightarrow{a} \times [\overrightarrow{a} \times (\overrightarrow{a} \times \overrightarrow{b})]$ 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

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