Write the Cartesian equation of the following line given in vector form: \(\overrightarrow { r } =2\hat { i } +\hat { j } +4\hat { k } +\lambda (\hat { i } +\hat { j } -\hat { k } )\)

Find the coordinates of a point, where the line \(\frac { x+2 }{ 1 } =\frac { y-5 }{ 3 } =\frac { z+1 }{ 5 } \) cuts yz-plane.

Write the equation of the straight line through the point \((\alpha ,\beta ,\gamma )\) and parallel to z-axis

Write the sum of intercepts cut off by the plane \(\overset { \rightarrow }{ r } =(2\overset { \wedge }{ i } +\overset { \wedge }{ j } -k)-5=0\) on the three axes

write the vector equation of the line passing through (1, 2, 3) and perpendicular to the plane \(\check { r } +(\check { r } +2\check { j } -5\check { k } )+9=0\)

Find the length of the perpendicular drawrt from the origin to the plane 2x - 3y + 6z + 21 = 0

Let \(I_{ e }m_{ i }n_{ i }i=1,2,3\) be the direction cosines of three mutually perpendicular vector ion space
\( \left[ \begin{matrix} { l }_{ 4 } & { m }_{ 1 } & { n }_{ 1 } \\ { l }_{ 2 } & { m }_{ 2 } & { n }_{ 2 } \\ { l }_{ 3 } & { m }_{ 3 } & { n }_{ 3 } \end{matrix} \right] \)
Show that AA^'= l₃, where A = 

Find the distance between the point (5, ,4, - 6) and its image in xy-plane.

Show that the line through the points (0, 3, 2), (3, 5, 6) is perpendicular of the line through the points (1, - 1, 2) and (3,4, - 2).

Find the angle between the planes 7x + 2y + 6z = 15 and 3x-y + 10z = 17.

Find the angle between line \(\frac { x-1 }{ 6 } =\frac { y+3 }{ 2 } =\frac { z-2 }{ 3 } \) and the plane 2x - Y + 2z - 13 = 0.

Find the distance of the point(-1,-5,-10) from the point of intersection of the line:
\(\vec { r } .\left( \hat { i } -\hat { j } +2\hat { k } \right) +\lambda \left( 3\hat { i } +4\hat { j } +2\hat { k } \right) \) and the plane \(\vec { r } .\left( \hat { i } -\hat { j } +\hat { k } \right) =5\)

Find the shortest distance between the lines
\(\vec { r } =\hat { (i } +2\hat { j } +\hat { k } )+\lambda (\hat { i } -\hat { j } +\hat { k } )and\)
\(\vec { r } =(2\hat { i } -\hat { j } -\hat { k } )+\mu (2\hat { i } +\hat { j } +2\hat { k } )\)

Find the angle between the line \(\frac { x+1 }{ 2 } =\frac { 3y+5 }{ 9 } =\frac { 3-z }{ -6 } \) and the plane 10x + 2y - 11z = 3.

Show that the lines \(\frac { x+1 }{ 3 } =\frac { y+3 }{ 5 } =\frac { z+5 }{ 7 } \) and \(\frac { x-2 }{ 1 } =\frac { y-4 }{ 3 } =\frac { z-6 }{ 5 } \) intersect each other. Find the point of intersection also.

Show that the lines \(\frac { x-a+d }{ \alpha -\delta } =\frac { y-a }{ \alpha } =\frac { z-a-d }{ \alpha +\delta } \) and \(\frac { x-b+c }{ \beta -\gamma } =\frac { y-b }{ \beta } =\frac { z-b-c }{ \beta +\gamma } \) are coplanar 

Find the vector and cartesian equations of the plane passing through the line of intersection of the planes. \(\vec { r } .\left( 2\hat { i } +2\hat { j } -3\hat { k } \right) =7,\quad \vec { r } .\left( 2\hat { i } +5\hat { j } +3\hat { k } \right) =9\)where x and z intercept are equal.

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 equation of the plane through the point (4, - 3, 2) and perpendicular to the line of intersection of the planes x - y + 2z - 3 = 0 and 2x - y - 3z = 0. Find the point of intersection of the line \(\vec { r } =\hat { i } +2\hat { j } -\hat { k } +\lambda \left( \hat { i } +3\hat { j } -9\hat { k } \right) \) and the plane obtained above.