If a line makes angles $\alpha, \beta, \gamma$ with the coordinate axes,then $\cos 2\alpha + \cos 2\beta + \cos 2\gamma = \dots$

$A$ line makes angles $\alpha, \beta, \gamma$ and $\delta$ with the diagonals of a cube. Prove that $\cos^{2} \alpha + \cos^{2} \beta + \cos^{2} \gamma + \cos^{2} \delta = \frac{4}{3}$.

If the projections of the line segment $\overrightarrow{AB}$ on $XY$,$YZ$,and $ZX$ planes are $\sqrt{15}$,$\sqrt{46}$,and $7$ respectively,then the projection of $\overrightarrow{AB}$ on the $y$-axis is

Find a vector $\vec{r}$ of magnitude $3 \sqrt{2}$ units which makes an angle of $\frac{\pi}{4}$ and $\frac{\pi}{2}$ with $y$ and $z$-axes,respectively.

The projections of a line on the coordinate axes are $2, 3, 6$. Then the length of the line is

Let $A(1,-1,2), B(6,11,2), C(1,2,6)$ be three points. If $l_1, m_1, n_1$ are the direction cosines of $AB$ and $l_2, m_2, n_2$ are the direction cosines of $AC$,then $|l_1 l_2+m_1 m_2+n_1 n_2|=$