$i \times (j \times k) = $

Let $\bar{a}=\hat{i}+\hat{j}$,$\bar{b}=2\hat{i}-\hat{k}$,and $\bar{c}=3\hat{i}-\hat{j}+\hat{k}$. Find the vector $\bar{p}$ that satisfies $\bar{p} \cdot \bar{a}=0$ and $\bar{p} \times \bar{b}=\bar{c} \times \bar{b}$.

If $\vec{a}=\hat{i}+\hat{j}+\hat{k}$,$\vec{c}=\hat{j}-\hat{k}$,$\vec{a} \times \vec{b}=\vec{c}$ and $\vec{a} \cdot \vec{b}=1$,then $\vec{b}$ is equal to:

Let $\bar{a}$ and $\bar{b}$ be two vectors such that $|\bar{a}|=1$,$|\bar{b}|=4$,and $\bar{a} \cdot \bar{b}=2$. If $\bar{c}=(2 \bar{a} \times \bar{b})-3 \bar{b}$,then the angle between $\bar{b}$ and $\bar{c}$ is

Let the lines $L_1: \frac{x + 1}{3} = \frac{y + 2}{1} = \frac{z + 1}{2}$ and $L_2: \frac{x - 2}{1} = \frac{y + 2}{2} = \frac{z - 3}{3}$. The unit vector perpendicular to both $L_1$ and $L_2$ is:

If $\bar{u}$ and $\bar{v}$ are two vectors represented in the following figure,then the value of $|\bar{u} \times \bar{v}|$ is: