Find a unit vector perpendicular to the vector $2\hat{i} - \hat{j} + 2\hat{k}$ and coplanar with the vectors $\hat{i} + 2\hat{j} - \hat{k}$ and $2\hat{i} + \hat{j} - \hat{k}$.

The moment of a force represented by $\overrightarrow{F} = i + 2j + 3k$ about the point $P(2, -1, 1)$ is given by the cross product of the position vector $\overrightarrow{r}$ and the force vector $\overrightarrow{F}$. If the origin is $O(0, 0, 0)$,then $\overrightarrow{r} = 2i - j + k$. Calculate the moment $\overrightarrow{M} = \overrightarrow{r} \times \overrightarrow{F}$.

Let $\overrightarrow{a}=\hat{i}+2 \hat{j}+3 \hat{k}$ and $\overrightarrow{b}=\hat{i}-2 \hat{j}-3 \hat{k}$ be two vectors. If $A_1$ is the area of the quadrilateral having $\vec{a}, \vec{b}$ as its diagonals and $A_2$ is the area of the parallelogram having $\overrightarrow{a}, \overrightarrow{b}$ as its two adjacent sides,then $A_1 \cdot A_2=$

Let $\bar{a} = \alpha \hat{i} + 3 \hat{j} - \hat{k}$,$\bar{b} = 3 \hat{i} - \hat{j} + \beta \hat{k}$,and $\bar{c} = \hat{i} + 2 \hat{j} - 2 \hat{k}$ where $\alpha, \beta \in R$,be three vectors. If the projection of $\bar{a}$ on $\bar{c}$ is $\frac{10}{3}$ and $\bar{b} \times \bar{c} = -6 \hat{i} + 10 \hat{j} + 7 \hat{k}$,then the value of $(\alpha + \beta)$ is equal to

If $\vec{a}+2 \vec{b}+3 \vec{c}=\vec{0}$ and $(\vec{a} \times \vec{b})+(\vec{b} \times \vec{c})+(\vec{c} \times \vec{a})=\lambda(\vec{b} \times \vec{c})$,then the value of $\lambda$ is

If $a = 2i + 4j - 5k$ and $b = i + 2j + 3k$,then $|a \times b|$ is