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

Let $a, b, c$ be three unit vectors such that $a \times(b \times c)=\frac{1}{2} b$. If the angle between $a$ and $b$ is $\theta_1$ and the angle between $a$ and $c$ is $\theta_2$, then $\theta_1+\theta_2$ is equal to (in $^{\circ}$)

If $(\vec{a} \times \vec{b}) \times \vec{c} = \vec{a} \times (\vec{b} \times \vec{c})$ where $\vec{a}, \vec{b},$ and $\vec{c}$ are any three vectors such that $\vec{a} \cdot \vec{b} \neq 0$ and $\vec{b} \cdot \vec{c} \neq 0$,then $\vec{a}$ and $\vec{c}$ are:

If $\vec{a} = -\hat{i} + \hat{j} + \hat{k}$ and $\vec{b} = 2\hat{i} + 0\hat{j} + \hat{k}$,find a vector $\vec{c}$ satisfying the following conditions:
$(i)$ $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$.
$(ii)$ $\vec{c}$ is perpendicular to $\vec{b}$.
$(iii)$ $\vec{a} \cdot \vec{c} = 7$.

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:

Which of the following is a true statement?