The number of vectors of unit length perpendicular to the two vectors $a=(1,1,0)$ and $b=(0,1,1)$ is

The area of the parallelogram,whose diagonals are $\vec{d}_1 = 2 \hat{i} - \hat{j} + \hat{k}$ and $\vec{d}_2 = \hat{i} + 3 \hat{j} - \hat{k}$,is equal to

Let $\overline{a}, \overline{b}, \overline{c}$ be three vectors such that $|\overline{a}|=\sqrt{3}$,$|\overline{b}|=5$,$\overline{b} \cdot \overline{c}=10$ and the angle between $\overline{b}$ and $\overline{c}$ is $\frac{\pi}{3}$. If $\overline{a}$ is perpendicular to the vector $\overline{b} \times \overline{c}$,then $|\overline{a} \times(\overline{b} \times \overline{c})|$ is equal to

If $\vec{a}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=\hat{j}-\hat{k},$ find a vector $\vec{c}$ such that $\vec{a} \times \vec{c}=\vec{b}$ and $\vec{a} \cdot \vec{c}=3.$

If $\bar{a}=\hat{j}-\hat{k}$ and $\bar{c}=\hat{i}-\hat{j}-\hat{k}$,then the vector $\bar{b}$ satisfying $\bar{a} \times \bar{b}+\bar{c}=\vec{0}$ and $\bar{a} \cdot \bar{b}=3$ is

If the diagonals of a parallelogram are represented by the vectors $3\hat{i} + \hat{j} - 2\hat{k}$ and $\hat{i} + 3\hat{j} - 4\hat{k}$,then its area in square units is: