If the volume of the tetrahedron whose coterminous edges are given by the vectors $\bar{a} = -2\hat{i} + 3\hat{j} - 3\hat{k}$, $\bar{b} = 4\hat{i} + 5\hat{j} + (\lambda - 10)\hat{k}$, $\bar{c} = 6\hat{i} + 2\hat{j} - 3\hat{k}$ is 11 cubic units, then the sum of the possible values of $\lambda$ is $\ldots$

The value of $\theta \in \left(0, \dfrac{\pi}{2}\right)$ for which vectors $\bar{a} = (\sin\theta)\hat{i} + (\cos\theta)\hat{j}$ and $\bar{b} = \hat{i} - \sqrt{3}\hat{j} + 2\hat{k}$ are perpendicular is

If $|\bar{a}| = |\bar{b}| = 1, |\bar{c}| = 2$ and $\bar{a} \times (\bar{a} \times \bar{c}) + \bar{b} = \bar{0}$, then the acute angle between $\bar{a}$ and $\bar{c}$ is $\ldots$

The value of $|\bar{a} \cdot \bar{b}|^2 + |\bar{a} \times \bar{b}| \cdot |\bar{a} \times \bar{b}|$ is $\ldots$