$A$ line makes the same angle $\theta$ with each of the $x$ and $z$ axes. If the angle $\beta$ which it makes with the $y$-axis is such that $\sin^2 \beta = 3\sin^2 \theta$,then $\cos^2 \theta$ is-

If the direction cosines of two lines are such that $2l + m + 2n = 0$ and $3l^2 + 5m^2 - 11n^2 = 0$,then the angle between the two lines is

Let $A, B, C$ be three points on $\overline{OX}, \overline{OY}, \overline{OZ}$ respectively at distances $3, 6, 9$ from the origin $O(0, 0, 0)$. Let $Q$ be the point $(2, 5, 8)$ and $P$ be the point equidistant from $O, A, B, C$. Then,the coordinates of the point $R$ which divides $PQ$ in the ratio $3:2$ is

The angle between the lines,whose direction cosines $l, m, n$ satisfy the equations $l+m+n=0$ and $2l^2+2m^2-n^2=0$,is (in $^{\circ}$)

$A$ line passes through the points $A (6, -7, -1)$ and $B (2, -3, 1)$. Find the direction cosines of the line such that the angle $\alpha$ made with the $x$-axis is acute.

If a line makes an angle of $\frac{\pi}{4}$ with the positive directions of each of the $x$-axis and $y$-axis,then the angle that the line makes with the positive direction of the $z$-axis is