If the coordinates of the points $P, Q, R, S$ are $(1, 2, 3), (4, 5, 7), (-4, 3, -6)$ and $(2, 0, 2)$ respectively,then:

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 $l m+m n+n l$ takes the maximum value when

$A$ line $AB$ in three-dimensional space makes angles $45^{\circ}$ and $120^{\circ}$ with the positive $x$-axis and the positive $y$-axis respectively. If $AB$ makes an acute angle $\theta$ with the positive $z$-axis,then $\theta$ equals

If a vector $\vec{r}$ with direction cosines $l, m, n$ is equally inclined to the coordinate axes, then the total number of such vectors is

If the direction ratios of two lines are $a_1, b_1, c_1$ and $a_2, b_2, c_2$,then when are these lines parallel?

The direction cosines of the line which is perpendicular to the lines with direction cosines proportional to $\langle 1, -2, -2 \rangle$ and $\langle 0, 2, 1 \rangle$ is given by