The acute angle between the vector $2\hat{i} + \hat{j} - 3\hat{k}$ and the plane containing the vectors $2\hat{i} + 3\hat{j} - \hat{k}$ and $\hat{i} - \hat{j} + 2\hat{k}$ is

A parallelogram is constructed on $5\bar{a} + 2\bar{b}$ and $\bar{a} - 3\bar{b}$ as its adjacent sides, with $|\bar{a}| = 2\sqrt{2}, |\bar{b}| = 3$ . The angle between $\bar{a}$ and $\bar{b}$ is $\dfrac{\pi}{4}$ . Then the length of the diagonals of the parallelogram are

The lines $\dfrac{x - 2}{1} = \dfrac{y - 3}{1} = \dfrac{z - 4}{-k}$ and $\dfrac{x - 1}{k} = \dfrac{y - 4}{2} = \dfrac{z - 5}{1}$ are coplanar if

The direction ratios of the normal to the plane passing through (1, 0, 0), (0, 1, 0) which makes an angle of measure $45^\circ$ with the plane $2x + 3y = 7$ are....