If a line makes angles $\alpha, \beta, \gamma$ with the coordinate axes in three-dimensional space,then find the value of $\cos 2\alpha + \cos 2\beta + \cos 2\gamma$.

The direction cosines of a line are $\langle \frac{-9}{11}, \frac{6}{11}, \frac{-2}{11} \rangle$ respectively. Then its direction ratios are

$A$ line passes through the points $A(6, -7, -1)$ and $B(2, -3, 1)$. Find the direction cosines of the line such that the angle made by the line with the positive direction of the $x$-axis is acute.

If the direction cosines of a line are $\left(\frac{a}{\sqrt{83}}, \frac{5}{\sqrt{83}}, \frac{c}{\sqrt{83}}\right)$ and $c-a=4$,then $ca=$

If $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ are the direction cosines of two lines making an angle $\theta$ with each other,then $\cos \theta =$

If $\left( \frac{1}{2}, \frac{1}{3}, n \right)$ are the direction cosines of a line,then the value of $n$ is