If $a=(1,2,3), b=(2,-1,1), c=(3,2,1)$ and $a \times(b \times c)=\alpha a+\beta b+\gamma c$,then

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

Let $\overline{a}, \overline{b}$ and $\overline{c}$ be three unit vectors such that $\overline{a} \times(\overline{b} \times \overline{c})=\frac{\sqrt{3}}{2}(\overline{b}+\overline{c})$. If $\overline{b}$ is not parallel to $\overline{c}$,then the angle between $\overline{a}$ and $\overline{b}$ is

If $a = i + j - k$,$b = i - j + k$,and $c = i - j - k$,then $a \times (b \times c) = \dots$

If $\vec{a}$ and $\vec{b}$ are two non-collinear vectors, then $\frac{\vec{a} \times(\vec{b} \times \vec{a})}{|\vec{a}|^2}$ represents:

If $|\vec{a}| = |\vec{b}| = 1$, $|\vec{c}| = 2$ and $\vec{a} \times (\vec{a} \times \vec{c}) + \vec{b} = \vec{0}$, then the acute angle between $\vec{a}$ and $\vec{c}$ is ...