The value of $[a - b, b - c, c - a]$,where $|a| = 1, |b| = 5$ and $|c| = 3$,is

For how many distinct real values of $\lambda$ are the vectors $-\lambda^2 \hat{i} + \hat{j} + \hat{k}$,$\hat{i} - \lambda^2 \hat{j} + \hat{k}$,and $\hat{i} + \hat{j} - \lambda^2 \hat{k}$ coplanar?

$\bar{a}, \bar{b}, \bar{c}$ are three unit vectors such that $x \bar{a} + y \bar{b} + z \bar{c} = p(\bar{b} \times \bar{c}) + q(\bar{c} \times \bar{a}) + r(\bar{a} \times \bar{b})$. If $(\bar{a}, \bar{b}) = (\bar{b}, \bar{c}) = (\bar{c}, \bar{a}) = \frac{\pi}{3}$, $(\bar{a}, \bar{b} \times \bar{c}) = \frac{\pi}{6}$ and $\bar{a}, \bar{b}, \bar{c}$ form a right-handed system, then $\frac{x+y+z}{p+q+r} = $

If $\bar{a}, \bar{b},$ and $\bar{c}$ are mutually perpendicular vectors such that $|\bar{a}| = 1, |\bar{b}| = 3,$ and $|\bar{c}| = 5,$ then find the value of $[\bar{a} - 2\bar{b}, \bar{b} - 3\bar{c}, \bar{c} - 4\bar{a}]$.

If $\bar{x}=\frac{\bar{b} \times \bar{c}}{[\bar{a} \bar{b} \bar{c}]}, \bar{y}=\frac{\bar{c} \times \bar{a}}{[\bar{a} \bar{b} \bar{c}]}$ and $\bar{z}=\frac{\bar{a} \times \bar{b}}{[\bar{a} \bar{b} \bar{c}]}$ where $\bar{a}, \bar{b}, \bar{c}$ are non-coplanar vectors,then the value of $\bar{x} \cdot(\bar{a}+\bar{b})+\bar{y} \cdot(\bar{b}+\bar{c})+\bar{z} \cdot(\bar{c}+\bar{a})$ is

If $[(\overline{a}+2 \overline{b}+3 \overline{c}) \times(\overline{b}+2 \overline{c}+3 \overline{a})] \cdot(\overline{c}+2 \overline{a}+3 \overline{b})=54$,then the value of $[\overline{a} \ \overline{b} \ \overline{c}]$ is