If $a, b$ and $c$ are three vectors with magnitudes $1, 1$ and $2$ respectively and $a \times (a \times c) + b = 0$, then the angle between $a$ and $c$ is

Statement $(A)$ : If $\vec{a}$ is perpendicular to $\vec{b}$ and $\vec{c}$,then $\vec{a} \times (\vec{b} \times \vec{c}) = 0$.
Reason $(R)$ : If $\vec{b}$ is perpendicular to $\vec{c}$,then $\vec{b} \times \vec{c} = 0$.

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

Let $\vec{p}, \vec{q},$ and $\vec{r}$ be three non-coplanar unit vectors equally inclined to each other at an acute angle $\theta$. The value of $|\vec{p} \times (\vec{q} \times \vec{r})|$ is:

Let $\overline{a}=\hat{j}-\hat{k}$ and $\overline{c}=\hat{i}-\hat{j}-\hat{k}$. Then the vector $\overline{b}$ satisfying $\overline{a} \times \overline{b}+\overline{c}=\overline{0}$ and $\overline{a} \cdot \overline{b}=3$,is

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