$A$ line makes angles $\frac{\alpha}{2}, \frac{\beta}{2}, \frac{\gamma}{2}$ with the positive directions of the coordinate axes ($x, y, z$ axes respectively). Then,the value of $\cos \alpha + \cos \beta + \cos \gamma$ 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 $|l_1 l_2+m_1 m_2+n_1 n_2|=$

The angle between the lines whose direction ratios satisfy the equations $l+m+n=0$ and $l^2=m^2+n^2$ is

Suppose $(l_1, m_1, n_1)$ and $(l_2, m_2, n_2)$ are the direction cosines of two lines and $\theta$ is the angle between them,where $\cos \theta = \pm(l_1 l_2 + m_1 m_2 + n_1 n_2)$. Let $A=(1, -2, 3)$,$B=(3, 1, -3)$,and $C=(-3, 1, 3)$ be the vertices of $\triangle ABC$. Then,$\cos A =$

If the direction cosines of a line $L$ are $(pq, q, q)$ and the angle between the line $L$ and the positive direction of the $X$-axis is $\frac{\pi}{3}$,then $p^2 : q^2 =$

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