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

If $\vec{a}=2 \hat{i}+3 \hat{j}$,$\vec{b}=3 \hat{j}+4 \hat{k}$,and $\vec{c}=5 \hat{i}+4 \hat{k}$ are three vectors,then a vector which is perpendicular to $\vec{a}$ and $\vec{b} \times \vec{c}$ is

Let $\overline{a}=\hat{j}-\hat{k}$ and $\overline{c}=\hat{i}-\hat{j}-\hat{k}$. Then the vector $\overline{b}$ satisfying $\overline{a} \times \overline{b}+\overline{c}=\overline{0}$ and $\overline{a} \cdot \overline{b}=3$,is

If $\bar{x} \cdot \bar{y} = 0$,then $\bar{x} \times (\bar{x} \times \bar{y}) = \dots$ where $|\bar{x}| = 1$.

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) $a \times (b \times c) = (a \cdot c) b - (a \cdot b) c$

For any vector $r$, the expression $i \times(r \times i) + j \times(r \times j) + k \times(r \times k)$ is equal to: