If the line $\frac{2-x}{3}=\frac{3y-2}{4\lambda+1}=4-z$ makes a right angle with the line $\frac{x+3}{3\mu}=\frac{1-2y}{6}=\frac{5-z}{7}$,then $4\lambda+9\mu$ is equal to :

$A(1, -2, 1)$ and $B(2, -1, 2)$ are the end points of a line segment. If $D(\alpha, \beta, \gamma)$ is the foot of the perpendicular drawn from $C(1, 2, 3)$ to $AB$,then $\alpha^2 + \beta^2 + \gamma^2 =$

Find the vector and the cartesian equations of the lines that passes through the origin and $(5, -2, 3).$

The shortest distance between the lines $\frac{x-4}{1} = \frac{y-3}{2} = \frac{z-2}{-3}$ and $\frac{x+2}{2} = \frac{y-6}{4} = \frac{z-5}{-5}$ is:

The position vector of the point of intersection of the line joining the points $2 \hat{i}-\hat{j}+6 \hat{k}$ and $3 \hat{i}-\hat{j}-7 \hat{k}$,and the line joining the points $2 \hat{i}+\hat{j}-6 \hat{k}$ and $3 \hat{i}-\hat{j}-7 \hat{k}$ is:

Find the coordinates of the point where the line passing through the points $A(3, 4, 1)$ and $B(5, 1, 6)$ crosses the $XY$-plane.