Find a vector that is perpendicular to both vectors $\hat{i} + \hat{j} + \hat{k}$ and $\hat{i} + \hat{j}$.

Let $\overrightarrow{a}=3\hat{i}-\hat{j}+2\hat{k}$,$\overrightarrow{b}=\overrightarrow{a}\times(\hat{i}-2\hat{k})$ and $\overrightarrow{c}=\overrightarrow{b}\times\hat{k}$. Then the projection of $\overrightarrow{c}-2\hat{j}$ on $\overrightarrow{a}$ is:

Prove that in a $\Delta ABC$,$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$,where $a, b, c$ represent the magnitudes of the sides opposite to vertices $A, B, C$,respectively.

If $a \times (b \times c) = 0,$ then

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

Let $\vec{a}=3 \hat{i}+\hat{j}-\hat{k}$ and $\vec{c}=2 \hat{i}-3 \hat{j}+3 \hat{k}$. If $\vec{b}$ is a vector such that $\vec{a}=\vec{b} \times \vec{c}$ and $|\vec{b}|^2=50$,then $|72-| \vec{b}+\vec{c}|^2 |$ is equal to $..........$.