Find the projection of the line segment joining the points $P(7, -5, 11)$ and $Q(-2, 8, 13)$ on a line $AB$ having direction cosines $\frac{1}{3}, \frac{2}{3}, \frac{2}{3}$.

Which of the following statements is false?

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 $\theta =$

If a line $L$ makes angles $\frac{\pi}{3}$ and $\frac{\pi}{4}$ with the positive $X$-axis and positive $Y$-axis respectively,then the angle made by $L$ with the positive direction of $Z$-axis is

The angle between the lines with direction ratios $(2, -2, 1)$ and $(1, -2, 2)$ is

Let the direction cosines of two lines satisfy the equations: $4l+m-n=0$ and $2mn+10nl+3lm=0$. Then the cosine of the acute angle between these lines is: