The projection of the line segment joining the points $(-1, 0, 3)$ and $(2, 5, 1)$ on the line whose direction ratios are $6, 2, 3$ is

If the angles made by a straight line with the coordinate axes are $\alpha, \frac{\pi}{2}-\alpha, \beta$, then $\beta$ is equal to

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 angle between the lines whose direction cosines are $\left(\frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{\sqrt{3}}{2}\right)$ and $\left(\frac{\sqrt{3}}{4}, \frac{1}{4}, \frac{-\sqrt{3}}{2}\right)$ is

If $\alpha, \beta, \gamma$ are the angles made by a line with $x, y,$ and $z$ axes respectively such that $2\left( \frac{\tan^2 \alpha}{1 + \tan^2 \alpha} + \frac{\tan^2 \beta}{1 + \tan^2 \beta} + \frac{\tan^2 \gamma}{1 + \tan^2 \gamma} \right) = 3 \sec^2 \frac{\theta}{2}$,then $\theta =$

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 $|l_1 l_2+m_1 m_2+n_1 n_2|=$