In three-dimensional space,a line $AB$ makes angles of $45^\circ$ and $120^\circ$ with the positive $x$-axis and $y$-axis,respectively. If $AB$ makes an acute angle $\theta$ with the positive $z$-axis,then $\theta = \dots^\circ$.

If a line makes angles of $30^o$ and $45^o$ with $X-$ axis and $Y-$ axis,then the angle made by it with $Z-$ axis is

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

What are the direction cosines of the line joining the points $(4, 3, -5)$ and $(-2, 1, -8)$?

The acute angle between the lines whose direction cosines are given by the equations $l+m+n=0$ and $2lm+2ln-mn=0$ is

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