If a line $L$ makes angles $\frac{\pi}{3}$ and $\frac{\pi}{4}$ with the positive $X$-axis and positive $Y$-axis respectively,then the angle made by $L$ with the positive direction of $Z$-axis is

If the direction cosines of a line are $\frac{1}{c}, \frac{1}{c}, \frac{1}{c}$,then:

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 $m_{1} n_{2}-m_{2} n_{1}, n_{1} l_{2}-n_{2} l_{1}, l_{1} m_{2}-l_{2} m_{1}$.

$A$ line makes angles $\alpha, \beta, \gamma$ with the coordinate axes. If $\alpha + \beta = 90^\circ$,then $\gamma = \dots \dots ^\circ$.

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 coordinates of the points $A, B, C, D$ are $(2, 3, -1), (3, 5, -3), (1, 2, 3)$ and $(3, 5, 7)$ respectively,then the projection of $AB$ on $CD$ is