If $\overline{a}$ and $\overline{c}$ are unit vectors inclined at $\frac{\pi}{3}$ with each other and $(\overline{a} \times (\overline{b} \times \overline{c})) \cdot (\overline{a} \times \overline{c}) = 5$,then the value of $5[\overline{a} \overline{b} \overline{c}] = $

If the vectors $2i - 3j$,$i + j - k$,and $3i - k$ form three concurrent edges of a parallelepiped,then the volume of the parallelepiped is

For what value of $a$ is the volume of the parallelepiped formed by the vectors $i + aj + k$,$j + ak$,and $ai + k$ minimum?

The vectors $i + 2j + 3k$,$\lambda i + 4j + 7k$,and $-3i - 2j - 5k$ are collinear if $\lambda$ equals:

$a \cdot (a \times b) = $

$\bar{a}, \bar{b}, \bar{c}$ 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} = $