Three Dimensional Geometry

The magnitude projection of line segment joining points $(1,2,3)$ and $(-1,4,2)$ on the line joining points $(-2,3,3)$ and $(0,6,-3)$ is

The equation of the line passing through \((-4,3,1)\) parallel to the plane \(x+2 y-z-5=0\) and intersecting the line \(\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z-2}{-1}\) is

If the plane \(3x + y + 2z + 6 = 0\) is parallel to the line \({{3x - 1} \over {2b}} = 3 - y = {{z - 1} \over a}\), then the value of \(3a + 3b\) is

The distance between the line \(r = 2\widehat i - 2\widehat j + 3\widehat k + \lambda (\widehat i - \widehat j + 4\widehat k)\) and the plane \(a\,.\,(\widehat i + 5\widehat j + \widehat k) = 5\) is

Consider the two lines
\({L_1}:{{x + 1} \over 3} = {{y + 2} \over 1} = {{z + 1} \over 2}\) and \({L_2}:{{x - 2} \over 1} = {{y + 2} \over 2} = {{z - 3} \over 3}\)
The unit vector perpendicular to both the lines L1 and L2 is

Angle between the diagonals of a cube is

A line passing through P(3, 7, 1) and R(2, 5, 7) meet the plane 3x + 2y + 11z \(-\) 9 = 0 at Q. Then PQ is equal to