If $u = i \times (a \times i) + j \times (a \times j) + k \times (a \times k),$ then

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

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

If $a, b, c$ are three unit vectors such that $a \times (b \times c) = \frac{\sqrt{3}}{2} b + \frac{1}{2} c$,then the angles between $a, b$ and $a, c$ respectively are

If $\vec{a}, \vec{b}, \vec{c}$ are three non-zero vectors and $\hat{n}$ is a unit vector perpendicular to $\vec{c}$ such that $\vec{a} = \alpha \vec{b} - \hat{n}, (\alpha \neq 0)$ and $\vec{b} \cdot \vec{c} = 12$,then $|\vec{c} \times (\vec{a} \times \vec{b})|$ is equal to:

Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors having magnitudes $1, 1$ and $2$ respectively. If $\vec{a} \times (\vec{a} \times \vec{c}) + \vec{b} = \vec{0}$, then the acute angle between $\vec{a}$ and $\vec{c}$ is