Find the vector and the Cartesian equations of a line passing through the point (1,2,-4) and parallel to the line joining the points A(3,3,-5) and B(1,0,-11). Hence, find the distance between the two lines. OR Find the equations of the line passing through the points A(1,2,3) and B(3,5,9). Hence, find the coordinates of the points on this line which are at a distance of 14 units from point B.

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 $\beta$.

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:

If the lines $\frac{x-1}{-3}=\frac{y-2}{2k}=\frac{z-3}{2}$ and $\frac{x-1}{3k}=\frac{y-1}{1}=\frac{z-6}{-7}$ are perpendicular to each other, find the value of k and hence write the vector equation of a line perpendicular to these two lines and passing through the point (3, -4, 7).

Find the coordinates of the foot of the perpendicular drawn from the point $P(0, 2, 3)$ to the line:$$\frac{x+3}{5} = \frac{y-1}{2} = \frac{z+4}{3}$$ OR (b) Three vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ satisfy the condition $\vec{a} + \vec{b} + \vec{c} = \vec{0}$. Evaluate the quantity $\mu = \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}$, if $|\vec{a}| = 3$, $|\vec{b}| = 4$, and $|\vec{c}| = 2$.