For any three vectors $a, b, c$,the condition $a \times (b \times c) = (a \times b) \times c$ holds if:

Let $\vec{v}$ be a unit vector which follows the equation $\vec{v} \times \vec{b} = \vec{c}$. Also,$|\vec{b}| = 2$ and $|\vec{c}| = \sqrt{3}$. Then,which of the following is true?

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 $\vec{b} \times \vec{c} = 0$.

If $a = i + j - 2k$, then $\sum \{(a \times i) \times j\}^2$ is equal to

If $\vec{a}, \vec{b},$ and $\vec{c}$ are vectors such that $|\vec{b}| = |\vec{c}|$,then $[(\vec{a} + \vec{b}) \times (\vec{a} \times \vec{c})] \times (\vec{b} \times \vec{c}) \cdot (\vec{b} + \vec{c}) = ...$

Let $\vec{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \vec{b}=2 \hat{i}-2 \hat{j}-2 \hat{k}$ and $\vec{c}=-\hat{i}+4 \hat{j}+3 \hat{k}$. If $\vec{d}$ is a vector perpendicular to both $\vec{b}$ and $\vec{c}$ and $\vec{a} \cdot \vec{d}=18$,then $|\vec{a} \times \vec{d}|^2$ is equal to $..........$.