Find the non-parametric form of vector equation of the plane passing through the point (1, −2, 4) and perpendicular to the plane x + 2y −3z = 11 and parallel to the line \(\frac { x+7 }{ 3 } =\frac { y+3 }{ -1 } =\frac { z }{ 1 } \)

Find the parametric form of vector equation and Cartesian equations of the plane containing the line \(\vec { r } =(\hat { i } -\hat { j } +3\hat { k } )+t(2\hat { i } -\hat { j } +4\hat { k } )\) and perpendicular to plane \(\vec { r } .(\hat { i } +2\hat { j } +\hat { k } )=8\)

Show that the lines \(\frac { x-2 }{ 1 } =\frac { y-3 }{ 1 } =\frac { z-4 }{ 3 } \) and \(\frac{x-1}{-3}=\frac{y-4}{2}=\frac{z-5}{1}\) coplanar. Also, find the plane containing these lines.

If the straight lines \(\frac { x-1 }{ 2 } =\frac { y+1 }{ \lambda } =\frac { z }{ 2 } \) and \(\frac { x-1 }{ 2 } =\frac { y+1 }{ \lambda } =\frac { z }{ \lambda } \) are coplanar, find λ and equations of the planes containing these two lines.

Find the image of the point whose position vector is \(\hat { i } +2\hat { j } +3\hat { k } \) in the plane \(\vec { r } .(\hat { i } +2\hat { j } +4\hat { k } )\) = 38