If $a \times b = b \times c \ne 0,$ where $a, b$ and $c$ are coplanar vectors,then for some scalar $k$

Let $\bar{a}, \bar{b}, \bar{c}$ be vectors such that $\bar{a} \neq \bar{o}, \bar{b} \neq \bar{o}, \bar{a} \times \bar{c} = \bar{b}$ and $\bar{b} \times \bar{c} = \bar{a}$. Then:

Find a unit vector perpendicular to each of the vectors $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b},$ where $\vec{a}=3 \hat{i}+2 \hat{j}+2 \hat{k}$ and $\vec{b}=\hat{i}+2 \hat{j}-2 \hat{k}.$

The area of a parallelogram whose diagonals are the vectors $2 \bar{a}-\bar{b}$ and $4 \bar{a}-5 \bar{b}$,where $\bar{a}$ and $\bar{b}$ are unit vectors forming an angle of $45^{\circ}$ is

If $\bar{a} = 4\hat{i} + 3\hat{j} + \hat{k}$ and $\bar{b} = \hat{i} - 2\hat{j} + 2\hat{k}$,then find the value of $\bar{a} \times (\bar{a} \times (\bar{a} \times (\bar{a} \times \bar{b})))$.

Let $\overrightarrow{a}=2 \hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{b}=\lambda \hat{j}+2 \hat{k}$, where $\lambda \in \mathbb{Z}$, be two vectors. Let $\overrightarrow{c}=\overrightarrow{a} \times \overrightarrow{b}$ and $\overrightarrow{d}$ be a vector of magnitude $2$ in the $yz$-plane. If $|\overrightarrow{c}|=\sqrt{53}$, then the maximum possible value of $(\overrightarrow{c} \cdot \overrightarrow{d})^2$ is equal to: