$A$ 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-

The angle between the lines whose direction cosines satisfy the equations $l + m + n = 0$ and $l^2 + m^2 - n^2 = 0$ is given by

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:

Consider the following statements:
Assertion $(A)$: The direction ratios of a line $L_1$ are $2, 5, 7$ and the direction ratios of another line $L_2$ are $\frac{4}{\sqrt{19}}, \frac{10}{\sqrt{19}}, \frac{14}{\sqrt{19}}$. Then the lines $L_1, L_2$ are parallel.
Reason $(R)$: If the direction ratios of a line $L_1$ are $a_1, b_1, c_1$, the direction ratios of a line $L_2$ are $a_2, b_2, c_2$ and $a_1 a_2 + b_1 b_2 + c_1 c_2 = 0$, then the lines $L_1, L_2$ are parallel. Which one of the following is true?

If the direction ratios of a line are proportional to $1, 2, 3$,find the projection of the line segment joining the points $(5, 2, 3)$ and $(-1, 0, 2)$ on the line.

If the line $\overrightarrow{OR}$ makes angles $\theta_{1}, \theta_{2}, \theta_{3}$ with the planes $XOY, YOZ, ZOX$ respectively,then $\cos ^{2} \theta_{1}+\cos ^{2} \theta_{2}+\cos ^{2} \theta_{3}$ is equal to