The direction cosines of the line which bisects the angle between the positive direction of $Y$ and $Z$ axes are

If $(l_1, m_1, n_1)$ and $(l_2, m_2, n_2)$ are the direction cosines of two lines,then $(l_1 m_2 - l_2 m_1)^2 + (m_1 n_2 - m_2 n_1)^2 + (n_1 l_2 - n_2 l_1)^2 + (l_1 l_2 + m_1 m_2 + n_1 n_2)^2 =$

If the direction ratios of a line are $1, 1, 2$,find the direction cosines of the line.

If $\theta$ is the angle between the lines whose direction cosines $(l, m, n)$ satisfy the equations $6mn - 2nl + 5lm = 0$ and $3l + m + 5n = 0$,then $\sin \theta = $

Each of the angles $\beta$ and $\gamma$ that a given line makes with the positive $y-$ and $z-$axes,respectively,is half of the angle that this line makes with the positive $x-$axis. Then the sum of all possible values of the angle $\beta$ is

The direction cosines $l, m, n$ of two lines satisfy the equations $3l + m + 5n = 0$ and $6mn - 2nl + 5lm = 0$. If $\theta$ is the angle between these lines, then $|\cos \theta| = $