Find the unit vector perpendicular to both vectors $\vec{a}$ and $\vec{b}$.

Let the vectors $\vec{a}, \vec{b}, \vec{c}$ be given as $\vec{a} = a_{1} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k}$,$\vec{b} = b_{1} \hat{i}+b_{2} \hat{j}+b_{3} \hat{k}$,and $\vec{c} = c_{1} \hat{i}+c_{2} \hat{j}+c_{3} \hat{k}$. Then show that $\vec{a} \times(\vec{b}+\vec{c})=\vec{a} \times \vec{b}+\vec{a} \times \vec{c}$.

Let $\bar{u}=\hat{i}+\hat{j}$,$\bar{v}=\hat{i}-\hat{j}$ and $\bar{w}=\hat{i}+2\hat{j}+3\hat{k}$. If $\hat{n}$ is a unit vector such that $\bar{u} \cdot \hat{n}=0$ and $\bar{v} \cdot \hat{n}=0$,then $|\bar{w} \cdot \hat{n}|$ is equal to

Given $\bar{a} = 2\bar{i} + \bar{j} - 2\bar{k}$ and $\bar{b} = \bar{i} + \bar{j}$. If $\bar{c}$ is a vector such that $\bar{a} \cdot \bar{c} = |\bar{c}|$,$|\bar{c} - \bar{a}| = 2\sqrt{2}$,and the angle between $\bar{a} \times \bar{b}$ and $\bar{c}$ is $30^{\circ}$,then the value of $|(\bar{a} \times \bar{b}) \times \bar{c}|^2$ is

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}| = 3$ and $|\vec{b}| = \frac{\sqrt{2}}{3}$. For what angle $\theta$ between $\vec{a}$ and $\vec{b}$ is $\vec{a} \times \vec{b}$ a unit vector?

If $\vec{a}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=2\hat{i}+\hat{j}+\hat{k}$ are two vectors,and $\vec{c}$ is a unit vector lying in the plane of $\vec{a}$ and $\vec{b}$ such that $\vec{c}$ is perpendicular to $\vec{b}$,then find the value of $\vec{c} \cdot (\hat{i}+\hat{j}+2\hat{k})$.