If a line makes an angle of \(\frac{\pi}{4}\) with the positive directions of both x-axis and z-axis, then the angle which it makes with the positive direction of y-axis is:

The Cartesian equation of the line passing through the point (1, -3, 2) and parallel to the line: \(\vec{r}=(2+\lambda)\hat{i}+\lambda\hat{j}+(2\lambda-1)\hat{k}\) is:

Position vectors of the points A, B and C as shown in the figure below are a, $\vec{b}$ and $\vec{c}$ respectively. If $\vec{AC}=\frac{5}{4}\vec{AB}$ , express $\vec{c}$ in terms of $\vec{a}$ and $\vec{b}$ . OR Check whether the lines given by equations $x=2\lambda+2$, $y=7\lambda+1$, $z=-3\lambda-3$ and $x=-\mu-2,$ $y=2\mu+8,$ $z=4\mu+5$ are perpendicular to each other or not.

If the angle between the lines $\frac{x-5}{\alpha}=\frac{y+2}{-5}=\frac{z+\frac{24}{5}}{\beta}$ and $\frac{x}{1}=\frac{y}{0}=\frac{z}{1}$ is $\frac{\pi}{4}$, find the relation between $\alpha$ and $\beta$.

The angle between the lines 2x=3y=-z and 6x=-y=-4z is