If a unit vector makes angles $\frac{\pi}{4}$ with $\hat{i}$, $\frac{\pi}{3}$ with $\hat{j}$ and $\theta \in (0, \pi)$ with $\hat{k}$, then a value of $\theta$ is equal to...

Find the direction cosines and the length of a vector whose projections on the coordinate axes are $6, -3, 2$.

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 $\delta \theta^{2}=\delta l^{2}+\delta m^{2}+\delta n^{2}$.

What do the three lines with direction ratios $(1, 1, 2)$,$(\sqrt{3}-1, -\sqrt{3}-1, 4)$,and $(-\sqrt{3}-1, \sqrt{3}-1, 4)$ represent?

If a line makes angles $\alpha, \beta, \gamma$ and $\delta$ with the four diagonals of a cube,then the value of $\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma + \sin^2 \delta$ is

If a line makes angles $90^{\circ}$,$135^{\circ}$,and $45^{\circ}$ with the positive directions of $X$,$Y$,and $Z$-axes respectively,then its direction cosines are: