Direction Cosines and Direction Ratios

The direction cosines of the line passing through $P\left(2, 3, -1\right)$ and the origin are

The direction cosines of the line making angles $\frac{\pi }{4}, \frac{\pi }{3}$ and $\theta \left(0<\theta <\frac{\pi }{2}\right)$ respectively with $x, y$ and $z$ axes, are

The lines whose direction cosines are given by the relations $al+bm+cn=0$ and $mn+nl+lm=0$ are

If $l, m, n$ are the direction cosines of a line which makes angles $\alpha , \beta$ and $\gamma$ with the coordinate axes $X, Y, Z$, respectively, then $lm+mn+nl$ takes the maximum value when