Which of the following is a true statement?

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

Let $\vec{a} = \hat{j} - \hat{k}$ and $\vec{c} = \hat{i} - \hat{j} - \hat{k}$. If $\vec{a} \times \vec{b} + \vec{c} = \vec{0}$ and $\vec{a} \cdot \vec{b} = 3$,find the vector $\vec{b}$.

Let $\overline{a}, \overline{b}$ and $\overline{c}$ be three non-zero vectors such that no two of them are collinear and $(\overline{a} \times \overline{b}) \times \overline{c} = \frac{1}{3}|\overline{b}||\overline{c}| \overline{a}$. If $\theta$ is the angle between vectors $\overline{b}$ and $\overline{c}$,then the value of $\operatorname{cosec} \theta$ is

Let $\vec{a}, \vec{b},$ and $\vec{c}$ be three unit vectors such that $\vec{a} \times (\vec{b} \times \vec{c}) = \frac{\sqrt{3}}{2}(\vec{b} + \vec{c})$. If $\vec{b}$ is not parallel to $\vec{c}$,then the angle between $\vec{a}$ and $\vec{b}$ is:

Let $\vec{a}=2 \hat{i}-\hat{j}+2 \hat{k}$ and $\vec{b}=\hat{i}+2 \hat{j}-\hat{k}$. Let a vector $\vec{v}$ be in the plane containing $\vec{a}$ and $\vec{b}$. If $\vec{v}$ is perpendicular to the vector $\vec{c}=3 \hat{i}+2 \hat{j}-\hat{k}$ and its projection on $\vec{a}$ is $19 \text{ units}$,then $|2 \vec{v}|^{2}$ is equal to .... .