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

The value of $x$ so that the volume of the parallelopiped formed by the vectors $\hat{i} + x\hat{j} + \hat{k}$, $\hat{j} + x\hat{k}$ and $x\hat{i} + \hat{k}$ is minimum, is

The sum of all real values of $\lambda$ for which the vectors $\vec{a} = \lambda\hat{i} + \hat{j} + \hat{k}$, $\vec{b} = \hat{i} + \lambda\hat{j} + 2\hat{k}$, $\vec{c} = 2\hat{i} + 3\hat{j} + \lambda\hat{k}$ are coplanar is...