The maximum volume of a parallelopiped (in cubic units) with vectors $(2a\hat{i} + \hat{k}), (a\hat{j} - a\hat{k})$, and $(3\hat{i} + a\hat{j})$, where $a \in [0, 1]$, as its coterminous edges is...

Let O$(0, 0)$, A$(-1, 2)$ and B$(1, 3)$ be the vertices of $\triangle$OAB. The bisector of angle O intersects side AB at point D. The value of $\vec{OD} \cdot \vec{AB}$ is equal to...

The perpendicular distance from the origin to the plane containing the points $(1, -2, 1), (2, -1, -3)$ and $(0, 1, 5)$ is...(in units)

If $\alpha, \beta, \gamma$ are the direction angles of the line $x = 4z + 3$ and $y = 2 - 3z$, then the value of $\cos\alpha + \cos\beta + \cos\gamma$ is...