The projections of a line on the coordinate axes are $4, 6, 12$. The direction cosines of the line are

$E(1,0,0), F(0,2,0), G(0,0,3)$ are respectively the mid-points of the sides $AB, BC, CA$ of $\triangle ABC$. If $a_1, b_1, c_1$ and $a_2, b_2, c_2$ are respectively the direction ratios of $AF$ and $BG$, then $\frac{a_1^2+b_1^2+c_1^2}{a_2^2+b_2^2+c_2^2}=$

The coordinates of $P \equiv(1, 2, 3)$ and $O \equiv(0, 0, 0)$ are given. Find the direction cosines of $\overline{OP}$.

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

If the direction cosines of two lines are given by $l+m+n=0$ and $l^2-5m^2+n^2=0$, then the angle between them is