Find the parametric form of vector equation of the straight line passing through (−1, 2,1) and parallel to the straight line \(\vec { r } =(2\hat { i } +3\hat { j } -\hat { k } )+t(\hat { i } -2\hat { j } +\hat { k } )\) and hence find the shortest distance between the lines.

Find the non-parametric form of vector equation, and Cartesian equation of the plane passing through the point (0, 1, -5) and parallel to the straight lines \(\vec { r } =(\hat { i } +2\hat { j } -4\hat { k } )+s(\hat { i } +3\hat { j } +6\hat { k } )\) and \(\hat { r } =(\hat { i } -3\hat { j } +5\hat { k } )+t(\hat { i } +\hat { j } -\hat { k } )\)

Find the non-parametric form of vector equation, and Cartesian equations of the plane \(\vec { r } =(6\hat { i } -\hat { j } +\hat { k } )+s(-\hat { i } +2\hat { j } +\hat { k } )+(-5\hat { i } -4\hat { j } -5\hat { k } )\)

If the straight lines \(\frac { x-1 }{ 1 } =\frac { y-2 }{ 1 } =\frac { z-3 }{ { m }^{ 2 } } \) and \(\frac { x-3 }{ 1 } =\frac { y-2 }{ { m }^{ 2 } } =\frac { z-1 }{ 2 } \) are coplanar, find the distinct real values of m.

Find the equation of the plane passing through the line of intersection of the planes \(\vec { r } .(2\hat { i } -7\hat { j } +4\hat { k } )=3\) and 3x - 5y + 11 = 0, and the point (-2, 1, 3)

Find the equation of the plane passing through the line of intersection of the planes x + 2y + 3z = 2 and x - y + z = 3 and at a distance \(\frac { 2 }{ \sqrt { 3 } } \) from the point (3, 1, -1)

Find the equation of the plane which passes through the point (3, 4, -1) and is parallel to the plane 2x - 3y + 5z = 0. Also, find the distance between the two planes.

Find the distance of the point (5, -5, -10) from the point of intersection of a straight line passing through the points A (4, 1, 2) and B (7, 5, 4) with the plane x - y + z = 5

Find the equation of the plane passing through the intersection of the planes \(\vec { r } .(\hat { i } +\hat { j } +\hat { k } )+1=0\) and \(\vec { r } .(2\hat { i } -3\hat { j } +5\hat { k } )=2\) and the point (-1, 2, 1).

Find the coordinates of the point where the straight line \(\vec { r } =(2\hat { i } -\hat { j } +2\hat { k } )+t(3\hat { i } +4\hat { j } +2\hat { k } )\) intersects the plane x−y+z−5 = 0.