If ABCDEF is a regular hexagon and $\overline{AB} + \overline{AC} + \overline{AD} + \overline{AE} + \overline{AF} = p\overline{AD} = q\overline{AO}$, where O is the center of the hexagon, then the values of $p$ and $q$ respectively are

The vector $\bar{r}$ whose magnitude is $3\sqrt{2}$ units and which makes angles of $\dfrac{\pi}{4}$ and $\dfrac{\pi}{2}$ with the positive y- and z-axes respectively is....

Let $(\bar{p} \wedge \bar{q})$ denote the angle between $\bar{p}$ and $\bar{q}$. If $\bar{a} + \bar{b} + \bar{c} = \bar{0}, |\bar{a}| = 7, |\bar{b}| = 5$ and $|\bar{c}| = 3$ then (take $\pi = \dfrac{22}{7}$)

If the lines $\dfrac{x-5}{5m+2} = \dfrac{2-y}{5} = \dfrac{1-z}{-1}$ and $x = \dfrac{2y+1}{4m} = \dfrac{1-z}{-3}$ are perpendicular to each other, then the value of m is ...