Find the distance of the plane 3x - 4y + 12z = 3 from the origin.

Find the Cartesian equation of the line which passes through the point (-2, 4, -5) and is parallel to the line \(\frac { x+3 }{ 3 } =\frac { 4-y }{ 5 } =\frac { z+8 }{ 6 } \)

Find the distance between the planes 2x + 3y + 4z = 10 and 4x + 6y + 8z = 18.

Find the distance of point \(2\check { i } +\check { j } -\check { k } \) from the plane \(\overrightarrow{r}.(\hat{i}-\hat{2}j+4\hat{k})=9. \)

If three consecutive vertices of a parallelogram are (3, - 1, - 1), (5, - 4,0), (2,3, - 2), then the coordinates of the fourth vertex .

Find the cartesian and vector equation of the line which passes through the point (- 2, 4, - 5) and parallel to the line given by \(\frac { x+3 }{ 3 } =\frac { y-4 }{ 5 } =\frac { x-z }{ -6 } \)

Find the vector equation of the plane which is at a distance of 5 units from the origin and normal to the plane is \(3\check { i } -2\check { j } +6\check { k } \)

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.

A plane meets the co-ordinate axes in A, B, and C such that the centroid of \(\triangle\)ABC is the point \((\alpha,\beta,\gamma).\) Show that the equation of the plane is \({{x}\over{\alpha}}+{{y}\over{\beta}}+{{z}\over{\gamma}}=3.\)

Find the angle between the planes 3x - 6y + 2z = 7 and 2x + 2y - 2z = 5.

Find the distance of a point (2, 5, - 3) from the plane:
\(\hat { r } .(6\hat { i } -3\hat { j } +2\hat { k }) =4\)

Find the equation of the line, which passes through the point(1, 2, 3) and is parallel to the vector \(3\vec { i } +2\vec { j } -2\vec { k } \)

The cartesian equation of the line is \(\frac { x-5 }{ 3 } =\frac { y+4 }{ 7 } =\frac { z-6 }{ 2 } \). Write its vector form.

If O be the origin and the co-ordinates of P are (1,2,-3), then find the equation of the plane passing through P and perpendicular to OP.

Find the Cartesian equation of the plane passing through the points A(0, 0, 0) and B(3, -1, 2) and parallel to the line \(\frac { x-4 }{ 1 } =\frac { y+3 }{ -4 } =\frac { z+1 }{ 7 } \) .

Find the equation of the perpendicular drawn from the point (1, -2, 3) to the plane 2x - 3y + 4z + 9 = 0. Also find the coordinates of the foot of the perpendicular.

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 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).