If lines \(\frac { x-1 }{ 2 } =\frac { y+1 }{ 3 } =\frac { z-1 }{ 4 } \) and \(\frac { x-3 }{ 1 } =\frac { y-k }{ 32 } =\frac { z }{ 1 } \)intersect, then find the value of k and hence find the equation of plane containing these lines.

Find the distance of the point \(3\hat { i } -2\hat { j } +\hat { k } \) from the plane 3x + y - z + 2 = 0 measured parallel to the line \(\frac { x-1 }{ 2 } =\frac { y+2 }{ -3 } =\frac { z-1 }{ 1 } \) . Also, find the foot of the . Also, find the foot of the perpendicular from the given point upon the given plane.

Find the equation of plane passing through the line of intersection of the planes\(\vec { r } .\left( 2\hat { i } +3\hat { j } -\hat { k } \right) =-1\) and \(\vec { r } .\left( \hat { i } +\hat { j } -2\hat { k } \right) =0\)and passing through the point (3,- 2, -1). Also, find the angle between the two given planes

Find the distance between the point (7, 2, 4) and the plane determined by the points A(2, 5, - 3) B(- 2, - 3, 5) and C(5, 3, - 3).

Find the distance of the point (2, 12, 5) from the point of intersection of the lines \(\vec { r } =\left( 2\hat { i } -4\hat { j } +2\hat { k } \right) +\lambda \left( 3\hat { i } +4\hat { j } +2\hat { k } \right) \) and the plane.

Find the distance of the point(3, 4, 5) from the plane x+y+z=2 measured parallel to the line 2x = y = z.

Find the foot of perpendicular from P(1, 2, -3) to the line\(\frac { x+1 }{ 2 } =\frac { y-3 }{ -2 } =\frac { z }{ -1 } \) . Also, find the image of P in the given line.

Find the position vector of foot of perpendicular and the perpendicular distance from the point P with position vector \(2\hat{i}+3\hat{j}+4\hat{k}\) to the plane \(\overrightarrow{r}.(2\hat{i}+\hat{j}+3\hat{k})-26=0.\) Also find the image of P in the plane.

Show that the lines \({{x+3}\over{-3}}={{y-1}\over{1}}={{z-5}\over{5}}\) and \({{x+1}\over{-1}}={{y-2}\over{2}}={{z-5}\over{5}}\) are coplanar. Also, find the equation of the plane containing these lines.

Find the equation of the plane passing through the line of intersection of the plane \(\overrightarrow{r}.(\hat{i}+3\hat{j})-6=0\) and \(\overrightarrow{r}.(3\hat{i}-\hat{j}-4\hat{k})=0,\) which is at a unit distance from the origin.