$A$ ray makes angles $\frac{\pi}{3}$ and $\frac{\pi}{4}$ with $Y$ and $Z$-axes respectively. Then,the value of the sine of the angle made by the ray with $X$-axis is

If a directed line makes angles $45^{\circ}$ and $60^{\circ}$ with the $X$ and $Y$-axes respectively,then the obtuse angle $\theta$ made by the line with the $Z$-axis is (in $^{\circ}$)

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 $\ell, m, n$ and $a, b, c$ are direction cosines of two lines,then:

The direction cosines of a line equally inclined to three mutually perpendicular lines having direction cosines as $(l_1, m_1, n_1)$,$(l_2, m_2, n_2)$ and $(l_3, m_3, n_3)$ are:

Let $O$ be the origin and $P$ be a point which is at a distance of $3$ units from the origin. If the direction ratios of $\vec{OP}$ are $(1, -2, -2)$,then the coordinates of $P$ are