Find the vector equation of the plane through the intersection of the planes \(\vec { r } .(\hat { i } +\hat { j } +\hat { k } )=6\) and \(\vec { r } .(2\hat { i } +3\hat { j } +4\hat { k } )=-5\) and the point (1, 1, 1)

Find the value of \(\lambda \) so that the lines \(\frac { 1-x }{ 3 } =\frac { 7y-14 }{ 2\lambda } =\frac { 5z-10 }{ 11 } \) and \(\frac { 7-7x }{ 3\lambda } =\frac { y-5 }{ 1 } =\frac { 6-z }{ 5 } \) are perpendicular to each other.

Find the equation of the plane passing through the point (-1, 3, 2) and perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0

Find the shortest distance between the following two lines:
\(\overrightarrow { r } =(1+\lambda )\acute { i } +(2-\lambda )\acute { j } +(\lambda +1)\acute { k } \\ \vec { r } =(2\acute { i } -\acute { j } -\acute { k } )+\mu (2\acute { i } +\acute { j } +2\acute { k } )\)

Find the vector and Cartesian equations of the line passing through (1, 2, -4) and perpendicular to the two lines : \(\frac { x-8 }{ 3 } =\frac { y+19 }{ -16 } =\frac { z-10 }{ 7 } \) and \(\frac { x-15 }{ 3 } =\frac { y-29 }{ 8 } =\frac { z-5 }{ -5 } \)

Find the equation of a line passing through the point P (2,-1,3) and perpendicular to the lines: \(\vec { r } =(\hat { i } +\hat { j } -\hat { k } )+\lambda (2\hat { i } -2\hat { j } +\hat { k } )\) and \(\vec { r } =(2\hat { i } -\hat { j } 3\hat { k } )+\mu (\hat { i } +2\hat { j } +2\hat { k } ).\)

If the lines \(\frac { x-1 }{ -3 } =\frac { y-2 }{ -2k } =\frac { z-3 }{ 2 } \)  and \(\frac { x-1 }{ k } =\frac { y-2 }{ 1 } =\frac { z-3 }{ 5 } \)are perpendicular, find the value of k and hence find the equation of plane containing these lines.

Find the vector and Cartesian equations of the line passing through the point P (1,2,3) and parallel to the planes \(\vec { r } .(\hat { i } -\hat { j } +2\hat { k } )=5\) and \(\vec { r } .(3\hat { i } +\hat { j } +\hat { k } )=6\).

Show that the lines :  \(\vec { r } =3\hat { i } +2\hat { j } -4\hat { k } +\lambda (\hat { i } +2\hat { j } +2\hat { k } )\) \(​​​​\vec { r } =5\hat { i } -2\hat { j } +\mu (3\hat { i } +2\hat { j } +6\hat { k } )\) Show that the lines are intersecting. Hence, find their point on intersection.

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 vector and cartesian forms of the equation of the plane passing through the point (1, 2, - 4) and parallel to the lines \(\\ \vec { r } =\left( \hat { i } +2\hat { j } -4\hat { k } \right) +\lambda \left( 2\hat { i } +3\hat { j } +6\hat { k } \right) \)
and \(\vec { r } =\left( \hat { i } -3\hat { j } +5\hat { k } \right) +\mu \left( \hat { i } +\hat { j } -\hat { k } \right) \)
Also, find the distance of the point (9, -8, -10) from the plane thus obtained.

Find the distance of the point (- 1,- 5, - 10) from the point of intersection of the line joining the points A(2, - 1, 2) and B(5, 3, 4) with the plane x - y + z = 5.

Find the distance of the point (1, - 2, 3) from the plane x - y + z = 5 measured parallel to the line \(\frac { x-1 }{ 2 } =\frac { y-3 }{ 3 } =\frac { z+2 }{ -6 } \)

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 co-ordinates of the point P where the line through A(3, - 4, - 5), 8(2, - 3, 1) crosses the plane, passing through the points (2, 2, 1), (3, 0, 1), (4, - 1, 0). Also, find the ratio in which P divides the line segment AB.

Find the equation of the plane which contains the line of intersection of the planes  \(\vec { r } .(\hat { i } -2\hat { j } +3\hat { k } )-4=0\) and  \(\vec { r } .(-2\hat { i } +\hat { j } +\hat { k } )+5=0\) and whose intercept on x-axis is equal to that of on y-axis.

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.

Find the vector equation of the plane that contains the lines \(\overrightarrow{r}=(\hat{i}+\hat{j})+\lambda(\hat{i}+2\hat{j}-\hat{k})\) and \(\vec{r}=(\hat{i}+\hat{j})+\mu(-\hat{i}+\hat{j}-2\hat{k}).\)
Also, find the length of perpendicular drawn from the point (2, 1, 4) to the plane thus obtained.