If a unit vector $\vec{r}$ makes angles $\frac{\pi}{3}$ with $\hat{i}$,$\frac{\pi}{4}$ with $\hat{j}$ and $\theta \in (0, \pi)$ with $\hat{k}$,then a value of $\theta$ is

Let $P(x, y, z)$ be a point in the first octant,whose projection in the $xy$-plane is the point $Q$. Let $OP = \gamma$; the angle between $OQ$ and the positive $x$-axis be $\theta$; and the angle between $OP$ and the positive $z$-axis be $\phi$,where $O$ is the origin. Then the distance of $P$ from the $x$-axis is:

The direction cosines of the line $x = y = z$ are

The direction cosines of the line making angles $\frac{\pi}{4}, \frac{\pi}{3}$ and $\theta$ $(0 < \theta < \frac{\pi}{2})$ respectively with $X, Y$ and $Z$ axes are:

The projection of the line segment joining the points $(2,1,-3)$ and $(-1,0,2)$ on the line whose direction ratios are $3,2,6$ is

If the direction cosines $(l, m, n)$ of two lines are connected by the relations $l+m+n=0$ and $lm=0$,then the angle between those lines is