$A$ unit vector perpendicular to both $i+j+k$ and $2i+j+3k$ is

The unit vector perpendicular to the vectors $6i + 2j + 3k$ and $3i - 6j - 2k$ is

The unit vector perpendicular to the vector $\hat{i}-2 \hat{j}+3 \hat{k}$ and coplanar with the vectors $\hat{i}+\hat{j}+\hat{k}$ and $2 \hat{i}-\hat{j}-\hat{k}$ is

Let $|\vec{a}|=2, |\vec{b}|=3$ and the angle between the vectors $\vec{a}$ and $\vec{b}$ be $\frac{\pi}{4}$. Then $|(\vec{a}+2 \vec{b}) \times(2 \vec{a}-3 \vec{b})|^2$ is equal to

If the area of the parallelogram with $\bar{a}$ and $\bar{b}$ as two adjacent sides is $15$ square units,then the area (in square units) of the parallelogram,having $3 \bar{a} + 2 \bar{b}$ and $\bar{a} + 3 \bar{b}$ as two adjacent sides,is

Let $\vec{a}=6 \hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=\hat{i}+\hat{j}$. If $\vec{c}$ is a vector such that $|\vec{c}| \geq 6$,$\vec{a} \cdot \vec{c}=6|\vec{c}|$,$|\vec{c}-\vec{a}|=2 \sqrt{2}$ and the angle between $\vec{a} \times \vec{b}$ and $\vec{c}$ is $60^{\circ}$,then $|(\vec{a} \times \vec{b}) \times \vec{c}|$ is equal to: