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

$a \times [a \times (a \times b)]$ is equal to

If $a, b, c$ are non-coplanar unit vectors such that $a \times (b \times c) = \frac{b + c}{\sqrt{2}}$,then the angle between $a$ and $b$ is

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 three unit vectors $a, b, c$ are such that $a \times (b \times c) = \frac{b}{2},$ then the vector $a$ makes with $b$ and $c$ respectively the angles

If $\bar{x} \cdot \bar{y} = 0$,then $\bar{x} \times (\bar{x} \times \bar{y}) = \dots$ where $|\bar{x}| = 1$.