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 $\alpha, \beta, \gamma, \delta$ with the four diagonals of a cube,then the value of ${\sin ^2}\alpha + {\sin ^2}\beta + {\sin ^2}\gamma + {\sin ^2}\delta$ is

Find the angle between two lines whose direction cosines are given by the relations $l + m + n = 0$ and $l^2 + m^2 - n^2 = 0$.

If $\alpha, \beta, \gamma$ are the angles made by a line with $x, y,$ and $z$ axes respectively such that $2\left( \frac{\tan^2 \alpha}{1 + \tan^2 \alpha} + \frac{\tan^2 \beta}{1 + \tan^2 \beta} + \frac{\tan^2 \gamma}{1 + \tan^2 \gamma} \right) = 3 \sec^2 \frac{\theta}{2}$,then $\theta =$

The direction ratios of the line joining the points $(4, 3, -5)$ and $(-2, 1, -8)$ are

The angle between a diagonal of a cube and one of its edges is ...................