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...

If $ABC$ is a right-angled triangle in which $BC$ is the longest side and the position vector of $B$ and $C$ are respectively $3\hat{i} - 2\hat{j} + \hat{k}$ and $5\hat{i} + \hat{j} - 3\hat{k}$, then the value of $\overline{AB} \cdot \overline{AC} + \overline{BA} \cdot \overline{BC} + \overline{CA} \cdot \overline{CB}$ is

The plane $\dfrac{x}{2} + \dfrac{y}{3} + \dfrac{z}{4} = 1$ cuts the axes at the points A, B, C then the area of triangle ABC is