Which of the following statements is true regarding the vector triple product $(a \times b) \times c$?

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 $\sin \theta$ is

Let $a, b, c$ be three vectors. Determine the correctness of the following statements:
$(i)$ $(a \times b) \times c = (a \cdot c) b - (b \cdot c) a$
(ii) $a \times (b \times c) = (a \cdot c) b - (a \cdot b) c$

Let $\vec{a}$ be a non-zero vector. If $\vec{x}=\hat{i} \times(\vec{a} \times \hat{i})$, $\vec{y}=\hat{j} \times(\vec{a} \times \hat{j})-\vec{a}$ and $\vec{z}=\hat{k} \times(\vec{a} \times \hat{k})-\vec{a}$, then $\left[\begin{array}{lll}\vec{x} & \vec{y} & \vec{z}\end{array}\right]=$

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:

The vectors $\bar{a}$ and $\bar{b}$ are not perpendicular and $\overline{c}$ and $\overline{d}$ are two vectors satisfying $\overline{b} \times \overline{c} = \overline{b} \times \overline{d}$ and $\overline{a} \cdot \overline{d} = 0$. Then the vector $\overline{d}$ is equal to: