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 vectors. Then $a \times (b \times c) = (a \times b) \times c$ if:

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 $a = 3i - j + 2k,$ $b = 2i + j - k$ and $c = i - 2j + 2k,$ then $(a \times b) \times c$ is equal to

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 $a, b, c, d$ are coplanar vectors,then $(a \times b) \times (c \times d) = $