Find the equation of the plane passing through each group of points:

(i) A(2, 2, -1), B(3, 4, 2) and C(7, 0, 6)

(ii) A(0, -1, -1), B(4, 5, 1) and C(3, 9, 4)

(iii) A(-2, 6, -6), B(-3, 10, 9) and

Show that the four points A(3, 2, -5),
B(-1, 4, -3), C(-3, 8, -5) and D(-3, 2, 1) are coplanar. Find the equation of the plane containing them.

Show that the four points A(0, -1, 0),
B(2, 1, -1), C(1, 1, 1) and D(3, 3, 0) are coplanar. Find the equation of the plane containing them.

Write the equation of the plane whose intercepts on the coordinate axes are 2, -4 and 5 respectively.

Reduce the equation of the plane 4x – 3y + 2z = 12 to the intercept form, and hence find the intercepts made by the plane with the coordinate axes.

Find the equation of the plane which passes through the point (2, -3,7) and makes equal intercepts on the coordinate axes.

A plane meets the coordinate axes at A, B and C respectively such that the centroid of ∆ABC is (1, -2, 3). Find the equation of the plane.

Find the Cartesian and vector equations of a plane passing through the point (1, 2, 3) and perpendicular to a line with direction ratios 2, 3, -4.

If O is the origin and P(1, 2, -3) be a given point, then find the equation of the plane passing through P and perpendicular to OP.

Find the vector and Cartesian equations of a plane which is at a distance of 5 units from the origin and which has as the unit vector normal to it.

Find the vector and Cartesian equations of a plane which is at a distance of 7 units from the origin and whose normal vector from the origin is

Find the vector and Cartesian equations of a plane which is at a distance of from the origin and whose normal vector from the origin is

Find the vector and Cartesian equations of a plane which is at a distance of 6 units from the origin and which has a normal with direction ratios 2, -1, -2.

Find the vector, and Cartesian equations of a plane which passes through the point (1, 4, 6) and the normal vector to the plane is

Find the length of the perpendicular from the origin to the plane Also write the unit normal vector from the origin to the plane.

Find the Cartesian equation of the plane whose vector equation is

Find the vector equation of a plane whose Cartesian equation is 5x - 7y + 2z + 4 = 0.

Find a unit vector normal to the plane
x – 2y + 2z = 6.

Find the direction cosines of the normal to the plane 3x – 6y + 2z = 7.

For each of the following planes, find the direction cosines of the normal to the plane and the distance of the plane from the origin:

(i) 2x + 3y - z = 5

(ii) z = 3

(iii) 3y + 5 = 0

Find the vector and Cartesian equations of the plane passing through the point (2, -1, 1) and perpendicular to the line having direction ratios 4, 2, -3.

Find the coordinates of the foot of the perpendicular drawn from the origin to the plane

(i) 2x + 3y + 4z -12 = 0

(ii) 5y + 8 = 0

Find the length and the foot of perpendicular drawn from the point (2, 3, 7) to the plane 3x – y – z = 7.

Find the length and the foot of the perpendicular drawn from the point (1, 1, 2) to the plane

From the point P(1, 2, 4), a perpendicular is drawn on the plane 2x + y - 2z + 3 = 0. Find the equation, the length and the coordinates of the foot of the perpendicular.

Find the coordinates of the foot of the perpendicular and the perpendicular distance from the point P( 3, 2, 1) to the plane 2x – y + z + 1 = 0.

Find also the image of the point P in the plane.

Find the coordinates of the image of the point P(1, 3, 4) in the plane 2x - y + z + 3 = 0.

Find the point where the line meets the plane 2x + 4y – z = 1.

Find the coordinates of the point where the line through (3, -4, -5) and (2, -3, 1) crosses the plane 2x + y + z = 7.

Find the distance of the point (2, 3, 4) from the plane 3x + 2y + 2z +5 =0, measured parallel to the line

Find the distance of the point (0, -3, 2) from the plane x + 2y -z = 1, measured parallel to the line

Find the equation of the line passing through the point P(4, 6, 2) and the point of intersection of the line and the plane x + y –Z = 8.

Show that the distance of the point of intersection of the line and the plane x -y + z = 5 from the point (-1, -5 -10) is 13 units.

Find the distance of the point A(-1, -5, -10) from the point of intersection of the line and the plane

HINT: Convert the equations of the line and the plane to Cartesian form.

Prove that the normals to the planes 4x + 11y + 2z + 3 = 0 and 3x - 2y + 5z = 8 are perpendicular to each other.

Show that the line is parallel to the plane

Find the equation of a plane which is at a distance of 3√3 units from the origin and the normal to which is equally inclined to the coordinate axes.

A vector of magnitude 8 units is inclined to the x-axis at 45^o, y-axis at 60^o and an acute angle with the z-axis, if a plane passes through a point (√2, -1, 1) and is normal to find its equation in vector form.

Find the vector equation of a line passing through the point and perpendicular to the plane

Also, find the point of intersection of this line and the plane.

Find the distance of the point from the plane

Find the distance of the point (3, 4, 5) from the plane

Find the distance of the point (1, 1, 2) from the plane

Find the distance of the point (2, 1, 0) from the plane 2x + y + 2z + 5 = 0.

Find the distance of the point (2, 1, - 1) from the plane x – 2y + 4z = 9.

Show that the point (1, 2, 1) is equidistant from the planes and

Show that the points ( - 3, 0, 1) and (1, 1, 1) are equidistant from the plane 3x + 4y – 12z + 13 = 0.

Find the distance between the parallel planes 2x + 3y + 4 = 4 and 4x + 6y + 8z = 12.

Find the distance between the parallel planes x + 2y - 2z + 4 = 0 and x + 2y – 2z – 8 = 0.

Find the equation of the planes parallel to the plane x – 2y + 2z – 3 = 0, each one of which is at a unit distance from the point (1, 1, 1).

Find the equation of the plane parallel to the plane 2x – 3y + 5z + 7 = 0 and passing through the point (3, 4, - 1). Also, find the distance between the two planes.

Find the equation of the plane mid - parallel to the planes 2x – 3y + 6z + 21 = 0 and 2x – 3y + 6z – 14 = 0

Show that the planes 2x – y + 6z = 5 and 5x – 2.5y + 15z = 12 are parallel.

Find the vector equation of the plane through the point and parallel to the plane

Find the vector equation of the plane passing through the point (a, b, b) and parallel to the plane

There is a error in question …… the point should be (a,b,c) instead of (a,b,b) to get the required answer.

Find the vector equation of the plane passing through the point (1, 1, 1) and parallel to the plane

Find the equation of the plane passing through the point (1, 4, - 2) and parallel to the plane 2x – y + 3z + 7 = 0.

Find the equations of the plane passing through the origin and parallel to the plane 2x – 3y + 7z + 13 = 0.

Find the equations of the plane passing through the point ( - 1, 0, 7) and parallel to the plane 3x – 5y + 4z = 11.

Find the equations of planes parallel to the plane x – 2y + 2z = 3 which are at a unit distance from the point (1, 2, 3).

Find the distance between the planes x + 2y + 3z + 7 = 0 and 2x + 4y + 6z + 7 = 0.

Find the equation of the plane through the line of intersection of the planes x + y + z = 6 and 2x + 2y + 4z + 5 = 0, and passing through the point (1, 1, 1).

Find the equation of the plane through the line of intersection of the planes x - 3y + z + 6 = 0 and x + 2y + 3z + 5 = 0, and passing through the origin.

Find the equation of the plane passing through the intersection of the planes 2x + 3y – z + 1 = 0 and x + y – 2z + 3 = 0, and perpendicular to the plane 3x - y -2z -4 = 0.

Find the equation of the plane passing through the line of intersection of the planes 2x - y = 0 and 3z - y = 0, and perpendicular to the plane 4x + 5y - 3z = 9.

Find the equation of the plane passing through the intersection of the planes x - 2y + z = 1 and 2x + y + z = 8, and parallel to the line with direction ratios 1, 2, 1. Also, find the perpendicular distance of (1, 1, 1) from the plane.

Find the equation of the plane passing through the line of intersection of the planes x + 2y + 3z – 5 = 0 and 3x - 2y –z + 1 = 0 and cutting off equal intercepts on the x-axis and z-axis.

Find the equation of the plane through the intersection of the planes 3x – 4y + 5z =10 and 2x + 2y - 3z = 4 and parallel to the line x = 2y = 3z.

Find the vector equation of the plane through the intersection of the planes and and passing through the point (2, 1, -1).

Find the vector equation of the plane through the point (1, 1, 1), and passing through the intersection of the planes and

Find the vector equation of the plane passing through the intersection of the planes and and passing through the point (-2, 1, 3).

Find the equation of the plane through the line of intersection of the planes and and perpendicular to the plane

Find the Cartesian and vector equations of the planes through the line of intersection of the planes and which are at a unit distance from the origin.

Find the acute angle between the following planes :

(i) and

(ii) and

(iii) and

(iv) and

Show that the following planes are at right angles:

(i) and

(ii) and

Find the value of λ for which the given planes are perpendicular to each other:

(i) and

(ii) and

Find the acute angle between the following planes:

(i) 2X – y + z = 5 and x + y + 2z = 7

(ii) x + 2y + 2z = 3 and 2x - 3y + 6z = 8

(iii) x + y - z = 4 and x + 2y + z = 9

(iv) x + y - 2z = 6 and 2x - 2y + z = 11

Show that each of the following pairs of planes are at right angles:

(i) 3x + 4y - 5z = 7 and 2x + 6y + 6z + 7 = 0

(ii) x - 2y + 4z = 10 and 18x + 17y + 4z = 49

Prove that the plane 2x + 2y + 4z = 9 is perpendicular to each of the planes x + 2y + 2z – 7 = 0 and 5x + 6y + 7z = 23.

Show that the planes 2x – 2y + 4z + 5 = 0 and 3x – 3y + 6z - 1 = 0 are parallel.

Find the value of λ for which the planes x – 4y + λz + 3 = 0 and 2x + 2y + 3z = 5 are perpendicular to each other.

Write the equation of the plane passing through the origin and parallel to the plane 5x - 3y + 7z + 11 = 0.

Find the equation of the plane passing through the point (a, b, c) and parallel to the plane

Find the equation of the plane passing through the point (1, -2, 7) and parallel to the plane 5x + 4y - 11z = 6.

Find the equation of the plane passing through the point A(-1, -1, 2) and perpendicular to each of the planes 3x + 2y - 3z = 1 and 5x – 4y + z = 5.

Find the equation of the plane passing through the origin and perpendicular to each of the planes x + 2y - z = 1 and 3x - 4y + z = 5.

Find the equation of the plane that contains the point A(1, -1, 2) and is perpendicular to both the planes 3x + 3y – 2z = 5 and x + 2y - 3z = 8. Hence, find the distance of the point P(-2, 5, 5) from the plane obtained above.

Find the equation of the plane passing through the points A(1, 1, 2) and B(2, -2, 2) and perpendicular to the plane 6x – 2y + 2z = 9.

Find the equation of the plane passing through the points A(-1, 1, 1) and B(1, -1, 1) and perpendicular to the plane x + 2y + 2z = 5.

Find the equation of the plane through the points A( 3, 4, 2) and B(7, 0, 6) and perpendicular to the plane 2x – 5y = 15.

HINT: The given plane is 2x – 5y + 0z = 15

Find the angle between the line and the plane

Find the angle between the line and the plane

Find the angle between the line and the plane 3x + 4y + z + 5 = 0.

Find the angle between the line and the plane 10x + 2y – 11z = 3.

Find the angle between the line joining the points A(3, - 4, - 2) and B(12, 2, 0) and the plane 3x – y + z = 1.

If the plane 2x – 3y – 6z = 13 makes an angle sin ^{- 1} (λ) with the x - axis, then find the value of λ.

Show that the line is parallel to the plane Also, find the distance between them.

Find the value of m for which the line is parallel to the plane

Find the vector equation of a line passing through the origin and perpendicular to the plane

Find the vector equation of the line passing through the point with position vector and perpendicular to the plane

Show that the equation ax + by + d = 0 represents a plane parallel to the z - axis. Hence, find the equation of a plane which is parallel to the z - axis and passes through the points A(2, - 3, 1) and B(- 4, 7, 6).

Find the equation of the plane passing through the points (1, 2, 3) and (0, - 1, 0) and parallel to the line

Find the equation of a plane passing through the point (2, - 1, 5), perpendicular to the plane x + 2y - 3z = 7 and parallel to the line

Find the equation of the plane passing through the intersection of the planes
5x - y + z = 10 and x + y - z = 4 and parallel to the line with direction ratios
2, 1, 1. Find also the perpendicular distance of (1, 1, 1) from this plane.

Find the vector and Cartesian equations of the plane passing through the origin and parallel to the vectors and

Find the vector and Cartesian equations of the plane passing through the point
(3, - 1, 2) and parallel to the lines and

Find the vector equation of a plane passing through the point (1, 2, 3) and parallel to the lines whose direction ratios are 1, - 1, - 2, and - 1, 0, 2.

Find the Cartesian and vector equations of a plane passing through the point (1, 2, - 4) and parallel to the lines and

Find the vector equation of the plane passing through the point and parallel to the vectors and

Find the vector and Cartesian forms of the equations of the plane containing the two lines and ..

Find the vector and Cartesian equations of a plane containing the two lines and Also show that the lines lies in the plane.

Prove that the lines and are coplanar. Also find the equation of the plane containing these lines.

Prove that the lines and are coplanar. Also find the equation of the plane containing these lines.

Show that the lines and are coplanar. Find the equation of the plane containing these lines.

Show that the lines and are coplanar. Find the equation of the plane containing these lines.

Show that the lines and are coplanar. Also find the equation of the plane containing these lines.

Find the equation of the plane which contains two parallel lines given by and

Find the direction ratios of the normal to the plane x + 2y - 3z = 5.

Find the direction cosines of the normal to the plane 2x + 3y - z = 4.

Find the direction cosines of the normal to the plane y = 3.

Find the direction cosines of the normal to the plane 3x + 4 = 0.

Write the equation of the plane parallel to XY-plane and passing through the point (4, -2, 3).

Write the equation of the plane parallel to YZ-plane and passing through the point
(-3, 2, 0).

Write the general equation of a plane parallel to the x-axis.

Write the intercept cut off by the plane 2x + y - z = 5 on the x-axis.

Write the intercepts made by the plane
4x - 3y + 2z = 12 on the coordinate axes.

Reduce the equation 2x – 3y + 5z + 4 = 0 to intercept form and find the intercepts made by it on the coordinate axes.

Find the equation of a plane passing through the points A(a, 0, 0), B(0, b, 0) and C(0, 0, c).

Write the value of k for which the planes 2x – 5y + kz = 4 and x + 2y – z = 6 are perpendicular to each other.

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

Find the angle between the planes and

Find the angle between the planes and

Find the angle between the line and the planes 10x + 2y – 11z = 3.

Find the angle between the line and the plane

Find the value of λ such that the line is perpendicular to the plane 3x - y – 2z = 7.

Write the equation of the plane passing through the point (a, b, c) and parallel to the plane

Find the length of perpendicular drawn from the origin to the plane 2x – 3y + 6z + 21 = 0.

Find the direction cosines of the perpendicular from the origin to the plane

Show that the line is parallel to the plane

Find the length of perpendicular from the origin to the plane

Find the value of λ for which the line

is parallel to the plane

Write the angle between the line

and the plane x + y + 4 = 0.

Write the equation of a plane passing through the point (2, -1, 1) and parallel to the plane 3x + 2y - z = 7.

Mark against the correct answer in each of the following:

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The direction cosines of the normal to the plane 5y + 4 = 0 are

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The length of perpendicular from the origin to the plane is

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The equation of a plane passing through the point A(2, -3, 7) and making equal intercepts on the axes, is

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A plane cuts off intercepts 3, -4, 6 on the coordinate axes. The length of perpendicular from the origin to this plane is

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If the line is parallel to the plane 2x – 3y + kz = 0, then the value of k is

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If O is the origin and P(1, 2, -3) is a given point, then the equation of the plane through P and perpendicular to OP is

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If the line lies in the plane 2x – 4y + z = 7, then the value of k is

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The plane 2x + 3y + 4z =12 meets the coordinate axes in A, B and C. The centroid of ∆ABC is

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If a plane meets the coordinate axes in A, B and C such that the centroid of ∆ABC is (1, 2, 4), then the equation of the plane is

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The equation of a plane through the point A(1, 0, -1) and perpendicular to the line is

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The line meets the plane 2x + 3y – z = 14 in the point

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The equation of the plane passing through the points A(2, 2, 1) and B(9, 3, 6) and perpendicular to the plane 2x + 6y + 6z = 1, is

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The equation of the plane passing through the intersection of the planes 3x - y + 2z – 4 = 0 and x + y + z - 2 = 0 and passing through the point A(2, 2, 1) is given by

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The equation of the plane passing through the points A(0, -1, 0), B(2, 1, -1) and C(1, 1, 1) is given by

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If the plane 2x – y + z = 0 is parallel to the line then the value of a is

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The angle between the line and a normal to the plane x – y + z = 0 is

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The point of intersection of the line and the plane 2x – y + 3z – 1 = 0, is

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The equation of a plane passing through the points A(a, 0, 0), B(0, b, 0) and C(0, 0, c) is given by

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If θ is the angle between the planes 2x – y + 2z = 3 and 6x – 2y + 3z = 5, then cos θ = ?

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The angle between the planes 2x – y + z = 6 and x + y + 2z = 7, is

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The angle between the planes and is

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The equation of the plane through the points A(2, 3, 1) and B(4, -5, 3), parallel to the x-axis, is

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A variable plane moves so that the sum of the reciprocals of its intercepts on the coordinate axes is (1/2). Then, the plane passes through the point

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The equation of a plane which is perpendicular to and at a distance of 5 units from the origin is

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The equation of the plane passing through the point A(2, 3,4) and parallel to the plane 5x - 6y + 7z = 3, is

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The foot of the perpendicular from the point A(7, 14, 5) to the plane 2x + 4y – z = 2 is

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The equation of the plane which makes with the coordinate axes, a triangle with centroid (α, β, γ) is given by

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The intercepts made by the plane are

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The angle between the line and the plane 2x – 3y + z = 5 is

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The angle between the line and the plane is

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The distance of the point from the plane is

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The distance between the parallel planes 2x – 3y + 6z = 5 and 6x – 9y + 18z + 20 = 0, is

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The distance between the planes x + 2y – 2z + 1 = 0 and 2x + 4y – 4z – 4z + 5 = 0, is

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The image of the point P(1, 3, 4) in the plane 2x – y + z + 3 = 0, is