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...

If $A(\vec{a})$, $B(\vec{b})$ and $C(\vec{c})$ are vertices of $\triangle ABC$. Point D divides segment BC internally in the ratio $2 : 1$. Point E divides segment AD internally in the ratio $1 : 2$, then the position vector of E is ____

If $|\vec{a}| = 3$, $|\vec{b}| = 4$, $|\vec{c}| = 5$ such that each vector is perpendicular to the sum of the other two, then $|\vec{a} + \vec{b} + \vec{c}|$ is equal to