If $|\bar{a}| = 4, |\bar{b}| = 3$ and $\bar{a} \cdot \bar{b} = 8$ then $[\bar{a}\ \ \bar{a} + \bar{b}\ \ \bar{a} \times \bar{b}] = $

Let $\bar{a}, \bar{b}, \bar{c}$ be the unit vectors such that $\bar{a}$ is perpendicular to $\bar{b}$ and the angle between $\bar{b}$ and $\bar{c}$ is $120^\circ$. If $\bar{a} + \bar{c}$ is perpendicular to $\bar{b} + \bar{c}$ then

Let $x_0$ be the point of local maxima of $f(x) = \bar{a} \cdot (\bar{b} \times \bar{c})$ where $\bar{a} = x\hat{i} - 2\hat{j} + 3\hat{k}, \bar{b} = -2\hat{i} + x\hat{j} - \hat{k}$ and $\bar{c} = 7\hat{i} - 2\hat{j} + x\hat{k}$ then the value of $\bar{a} \cdot \bar{c}$ at $x = x_0$ is

The parallelopiped is determined by vectors $\bar{a} = -2\hat{i} + 5\hat{j} + 3\hat{k}$, $\bar{b} = \hat{i} + 3\hat{j} - 2\hat{k}$, $\bar{c} = -3\hat{i} + \hat{j} + 4\hat{k}$. The altitude of parallelopiped on the parallelogram base determined by vectors $\bar{b}$ and $\bar{c}$ is