The projection of a line segment on the coordinate axes are $3, 4,$ and $5$ respectively. Find the length of the line segment.

If a line in space makes angles $\alpha, \beta$,and $\gamma$ with the coordinate axes,then $\cos 2\alpha + \cos 2\beta + \cos 2\gamma + \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma$ equals:

If a line makes angles $120^{\circ}$ and $60^{\circ}$ with the positive directions of $X$ and $Z$ axes respectively,then the angle made by the line with the positive $Y$-axis is (in $^{\circ}$)

$A$ point on the $XOZ$-plane divides the line segment joining the points $(5, -3, -2)$ and $(1, 2, -2)$ at:

If $l_{1}, m_{1}, n_{1}$ and $l_{2}, m_{2}, n_{2}$ are the direction cosines of two mutually perpendicular lines,show that the direction cosines of the line perpendicular to both of these are $m_{1} n_{2}-m_{2} n_{1}, n_{1} l_{2}-n_{2} l_{1}, l_{1} m_{2}-l_{2} m_{1}$.

If a variable line in two adjacent positions has direction cosines $l, m, n$ and $l+\delta l, m+\delta m, n+\delta n,$ show that the small angle $\delta \theta$ between the two positions is given by $\delta \theta^{2}=\delta l^{2}+\delta m^{2}+\delta n^{2}$.