The sine of the angle between the two vectors $3i + 2j - k$ and $12i + 5j - 5k$ will be

Let $\bar{a}, \bar{b}$ and $\bar{c}$ be three unit vectors such that $\bar{a} \times(\bar{b} \times \bar{c})=\frac{\sqrt{3}}{2}(\bar{b}+\bar{c})$. If $\bar{b}$ is not parallel to $\bar{c}$,then the angle between $\bar{a}$ and $\bar{b}$ is

The magnitude of a vector which is orthogonal to the vector $\hat{i}+\hat{j}+\hat{k}$ and is coplanar with the vectors $\hat{i}+\hat{j}+2\hat{k}$ and $\hat{i}+2\hat{j}+\hat{k}$ is

The area (in sq. units) of the parallelogram whose diagonals are along the vectors $8\hat{i} - 6\hat{j}$ and $3\hat{i} + 4\hat{j} - 12\hat{k}$ is:

If $a$ and $b$ are two non-zero perpendicular vectors,then a vector $y$ satisfying the equations $a \cdot y = c$ (where $c$ is a scalar) and $a \times y = b$ is

Let $\overrightarrow{a}=\hat{i}+5\hat{j}+\alpha\hat{k}$,$\overrightarrow{b}=\hat{i}+3\hat{j}+\beta\hat{k}$ and $\overrightarrow{c}=-\hat{i}+2\hat{j}-3\hat{k}$ be three vectors such that $|\overrightarrow{b} \times \overrightarrow{c}|=5\sqrt{3}$ and $\overrightarrow{a}$ is perpendicular to $\overrightarrow{b}$. Then the greatest value of $|\vec{a}|^{2}$ is .... .