Find the equation of the plane passing through the following points:

(2, 1, 0), (3, – 2, – 2), and (3, 1, 7)

Find the equation of the plane passing through the following points:

(– 5, 0, – 6), (– 3, 10, – 9) and (– 2, 6, – 6)

Find the equation of the plane passing through the following points:

(1, 1, 1), (1, – 1, 2) and (– 2, – 2, 2)

Find the equation of the plane passing through the following points:

(2, 3, 4), (– 3, 5, 1) and (4, – 1, 2)

Find the equation of the plane passing through the following points:

(0, – 1, 0), (3, 3, 0) and (1, 1, 1)

Show that the four point (0, – 1, – 1), (4, 5, 1), (3, 9, 4) and (– 4, 4, 4) are coplanar and find the equation of the common plane.

Show that the following points are coplanar.

(0, – 1, 0), (2, 1, – 1), (1, 1, 1) and (3, 3, 0)

Show that the following points are coplanar.

(0, 4, 3), (– 1, – 5, – 3), (– 2, – 2, 1) and (1, 1, – 1)

Find the coordinates of the point P where the line through A(3, – 4, – 5) and B(2, – 3, 1) crosses the plane passing through three points L(2, 2, 1), M(3, 0, 1) and N(4, – 1, 0). Also, find the ratio in which P divides the line segment AB.

Write the equation of the plane whose intercepts on the coordinate axes are 2, – 3 and 4.

Reduce the equations of the following planes in the intercept form and find its intercepts on the coordinate axes:

4x + 3y – 6z – 12 = 0

Reduce the equations of the following planes in the intercept form and find its intercepts on the coordinate axes:

2x + 3y – z = 6

Reduce the equations of the following planes in the intercept form and find its intercepts on the coordinate axes:

2x – y + z = 5

Find the equation of a plane which meets the axes in A, B and C, given that the centroid of the triangle ABC is the point (α, β, γ).

Find the equation of the plane passing through the point (2, 4, 6) and making equal intercepts of the coordinate axes.

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

Find the Cartesian form of the equation of a plane whose vector equation is

i.

ii.

Find the vector equation of the coordinates planes.

Find the vector equation of each one of the following planes:

i. 2x – y + 2z = 8

ii. x + y – z = 5

iii. x + y = 3

Find the vector and Cartesian equation of a plane passing through the point (1, -1, 1) and normal to the line joining the point (1, 2, 5) and (-1, 3, 1).

If is a vector of magnitude √3 is equally inclined with an acute with the coordinate axes. Find the vector and Cartesian forms of the equation of a plane which passes through (2, 1, -1) and is normal to .

The coordinates of the foot of the perpendicular drawn from the origin to a plane are (12, -4, 3). Find the equation of the plane.

Find the equation of the plane passing through the point (2, 3, 1) given that the direction ratios of normal to the plane are proportional to 5, 3, 2.

If the axes are rectangular and P is the point (2, 3, -1), find the equation of the plane through P at right angles to OP.

Find the intercepts made on the coordinate axes by the plane 2x + y – 2z = 3 and also find the direction cosines of the normal to the plane.

A plane passes through the point (1, -2, 5) and is perpendicular to the line joining the origin to the point . Find the vector and Cartesian forms of the equation of the plane.

Find the equation of the plane that bisects the line segment joining points (1, 2, 3) and (3, 4, 5) and is at right angle to it.

Show that the normal to the following pairs of planes are perpendicular to each other:

i. x – y + z – 2 =0 and 3x + 2y – z +4=0

ii. and

Show that the normal vector to the plane 2x + 2y + 2z = 3 is equally inclined with the coordinate axes.

Find a vector of magnitude 26 units normal to the plane 12x – 3y + 4z = 1.

If the line drawn from (4, -1, 2) meets a plane at right angles at the point (-10,5,4), find the equation of the plane.

Find the equation of the plane which bisects the line segment joining the points (-1, 2, 3) and (3, -5, 6) at right angles.

Find the vector and Cartesian equation of the plane which passes through the point (5, 2, -4) and perpendicular to the line with direction ratios 2, 3, -1.

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

If O is the origin and the coordinates of A are (a, b, c). Find the direction cosines of OA and the equation of the plane through A at right angles to OA.

Find the vector equation of the plane with intercepts 3, -4 and 2 on x, y and z axes respectively.

Find the vector equation of a plane which is at a distance of 3 units from the origin and has as the unit vector normal to it.

Reduce the equation 2x – 3y – 6z = 14 to the normal form and hence find the length of the perpendicular from the origin to the plane. Also, find the direction cosines of the normal to the plane.

Reduce the equation to normal form and hence find the length of perpendicular from the origin to the plane.

Write the normal form of the equation of the plane

2x–3y + 6z + 14 = 0.

The direction ratios of the perpendicular from the origin to a plane are 12, – 3, 4 and the length of the perpendicular is 5. Find the equation of the plane.

Find a normal unit vector to the plane x + 2y + 3z–6 = 0.

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 with the coordinate axes.

Find the equation of the plane passing through the point (1,2,1) and perpendicular to the line joining the points (1,4,2) and (2,3,5). Find also the perpendicular distance of the origin from this plane.

Find the vector equation of the plane which is at a distance of from the origin and its normal vector from the origin is . Also, find its Cartesian form.

Find the distance of the plane 2x – 3y + 4z – 6 = 0 from the origin.

Find the vector equation of the plane passing through the points (1, 1, 1), (1, – 1, 1) and (– 7, – 3, – 5).

Find the vector equation of the plane passing through the points P(2, 5, – 3), Q(– 2, – 3, 5) and R(5, 3, – 3).

Find the vector equation of the plane passing through point A(a, 0, 0), B(0, b, 0) and C(0, 0, c). Reduce it to normal form. If plane ABC is at a distance p from the origin, prove that .

Find the vector equation of the plane passing through the points (1, 1, – 1), (6, 4, – 5) and (– 4, – 2, 3).

Find the vector equation of the plane passing through the points , , and .

Find the angle between the planes :

Find the angle between the planes :

and

Find the angle between the planes :

and

Find the angle between the planes :

2x – y + z = 4 and x + y + 2z = 3

Find the angle between the planes :

x + y – 2z = 3 and 2x – 2y + z = 5

Find the angle between the planes :

x – y + z = 5 and x + 2y + z = 9

Find the angle between the planes :

2x – 3y + 4z = 1 and –x + y = 4

Find the angle between the planes :

2x + y – 2z = 5 and 3x – 6y – 2z = 7

Show that the following planes are at right angles :

and

Show that the following planes are at right angles :

x – 2y + 4z = 10 and 18x + 17y + 4z = 49

Determine the value of λ for which the following planes are perpendicular to each other.

and

Determine the value of λ for which the following planes are perpendicular to each other.

2x – 4y + 3z = 5 and x + 2y + λz = 5

Determine the value of λ for which the following planes are perpendicular to each other.

3x – 6y – 2z = 7 and 2x + y – λz = 5

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

Obtain the equation of the plane passing through the point (1, – 3, – 2) and perpendicular to the planes x + 2y + 2z = 5 and 3x + 3y + 2z = 8.

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 passing through the point (1, – 1, 2) and (2, – 2, 2) and which is perpendicular to the plane 6x – 2y + 2z = 9.

Find the equation of the plane passing through the points (2, 2, 1) and (9, 3, 6) and perpendicular to the plane 2x + 6y + 6z = 1.

Find the equation of the plane passing through the points whose coordinates are (– 1, 1, 1) and (1, – 1, 1) and perpendicular to the plane x + 2y + 2z = 5.

Find the equation of the plane with intercept 3 on the y - axis and parallel to ZOX plane.

Find the equation of the plane that contains the point (1, – 1, 2) and is perpendicular to each of the planes 2x + 3y–2z = 5and x + 2y – 3z = 8.

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

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

Find the vector equation of the plane through the points (2, 1, – 1) and (– 1, 3, 4) and perpendicular to the plane

x – 2y + 4z = 10.

Find the vector equation of the following planes in scalar product form

Find the vector equation of the following planes in scalar product form

Find the vector equation of the following planes in scalar product form

Find the vector equation of the following planes in scalar product form

Find the Cartesian form of the equation of the following planes :

Find the Cartesian form of the equation of the following planes :

Find the vector equation of the following planes in non – parametric form :

Find the vector equation of the following planes in non - parametric form :

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

Find the equation of the plane through (3, 4, –1) which is parallel to the plane

Find the equation of the plane through the intersection of the planes 2x – 7y + 4z – 3 = 0 and 3x – 5y + 4z + 11 = 0 and the point ( – 2,1,3)?

Find the equation of the plane through the point and passing through the line of intersection of the planes and

Find the equation of the plane through the line of intersection of the planes 2x – y = 0 and 3z – y = 0 which is perpendicular to the plane 4x + 5y – 3z = 8 ?

Find the equation of the plane through the line of intersection of the planes x + 2y + 3z – 4 = 0 and 2x + y – z + 5 = 0 which is perpendicular to the plane 5x + 3y – 6z + 8 = 0?

Find the equation of the plane through the line of intersection of the planes x + 2y + 3z + 4 = 0 and x – y + z + 3 = 0 and passing through the origin ?

Find the equation of the plane through the line of intersection of the planes x – 3y + 2z – 5 = 0 and 2x – y + 3z – 1 = 0 and passing through (1, – 2, 3)?

Find the equation of the plane through the line of intersection of the planes x – 3y + 2z – 5 = 0 and 2x – y + 3z – 1 = 0 which is perpendicular to the plane 5x + 3y – 6z + 8 = 0 ?

Find the equation of the plane through the line of intersection of the planes and , which is at a unit distance from the origin?

Find the equation of the plane through the line of intersection of the planes 2x + 3y – z + 1 = 0 and x + y – 2z + 3 = 0 which is perpendicular to the plane 3x – y – 2z – 4 = 0 ?

Find the vector equation of the plane passing through the intersection of the planes and = – 5 and the point (1, 1, 1).

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

Find the equation of family of planes through the line of intersection of the planes 3x – y + 2z = 4 and x + y + z = 2 which is passing through (2, 2, 1)?

Find the equation of family of planes through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0 and parallel to ?

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

Find the equation of the plane which contains the line of intersection of the plane x + 2y + 3z – 4 = 0 and 2x + y – z + 5 = 0 and whose x – intercept is twice of z – intercept. Hence, write the equation of the plane passing through the point (2,3, – 1) and parallel to the plane obtained above?

Find the equation of the plane through the line of intersection of the plane x + y + z = 1 and 2x + 3y + 4z = 5 and twice of its y–intercept is equals to the three times its z intercept?

Show that the points and are equidistant from the plane

Find the distance of the point (2, 3, –5) from the plane x + 2y – 2z – 9 = 0.

Find the equations of the planes parallel to the plane x + 2y – 2z + 8 = 0 which are at distance of 2 units from the point (2, 1, 1).

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

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

Find the distance of the point (2, 3, 5) from the xy–plane.

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

If the product of distances of the point (1, 1, 1) from the origin and the plane x – y + z + λ = 0 be 5, find the value of λ.

Find an equation for the set of all points that are equidistant from the planes 3x – 4y + 12z = 6 and 4x + 3z = 7.

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 (5, 3, –3).

A plane makes intercepts –6, 3, 4 respectively on the coordinate axes. Find the length of the perpendicular from the origin on it.

Find the distance of the point (1, –2, 4) from a plane passing through the point (1, 2, 2) and perpendicular to the planes x – y + 2z = 3 and 2x – 2y + z + 12 = 0.

Find the distance between the parallel planes 2x – y + 3z – 4 = 0 and 6x – 3y + 9z + 13 = 0.

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

Find the equation of the plane mid–parallel to the planes 2x – 2y + z + 3 = 0 and 2x – 2y + z + 9 = 0.

Find the distance between the planes and + 7 = 0.

Find the angle between the line and the plane 2x + y – z = 4.

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

The line is parallel to the plane Find m.

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

Find the vector equation of the line through the origin which is perpendicular to the plane

Find the equation of the plane through (2, 3, – 4) and (1, – 1, 3) and parallel to the x – axis.

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

Find the vector and Cartesian equations of the line passing through (1, 2, 3) and parallel to the planes and

Prove that the line of section of the planes 5x + 2y – 4z + 2 = 0 and 2x + 8y + 2z – 1 = 0 parallel to the plane 4x – 2y – 5z – 2 = 0.

Find the vector equation of the line passing through the point (1, – 1, 2) and perpendicular to the plane 2x – y + 3z – 5 = 0.

Find the equation of the plane through the points (2, 2, – 1) and (3, 4, 2) and parallel to the line whose direction ratios are 7, 0, 6.

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

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 proportional to 1, 2, 1. Find also the perpendicular distance of (1, 1, 1) from this plane.

State when the line is parallel to the plane Show that the line is parallel to the plane Also, find the distance between the line and the plane.

Show that the plane whose vector equation is and the line whose vector equation is are parallel. Also, find the distance between them.

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 and Cartesian forms of the equation of the plane passing through the point (1, 2, – 4) and parallel to the lines and Also, find the distance of the point (9, – 8, – 10) from the plane thus obtained.

Find the equation of the plane passing through the point (3, 4, 1) and (0, 1, 0) and parallel to the line

Find the coordinates of the point where the line intersects the plane x – y + z – 5 = 0. Also find the angle between the line and the plane.

Find the vector equation of the line passing through (1, 2, 3) and perpendicular to the plane

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

Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes and

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

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

Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the(a) yz –plane(b) zx –planeAlso, Find the angle which this line makes with these planes?

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, 12, 5) from the point of intersection of the line and the plane

Find the distance of the point P(– 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 P (3, 4, 4) from the point, where the line joining the points A(3, – 4, – 5) and B(2, – 3, 1) intersects the plane 2x + y + z = 7?

Find the distance of the point (1, – 5, 9) from the plane x – y + z = 5 measured along the line x = y = z ?

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

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

Find the equation of the plane containing the line and the point (0, 7, –7) and show that the line also lies in the same plane.

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

Show that the lines and 3x – 2y + z + 5 = 0 = 2x + 3y + 4z – 4 intersect. Find the equation of the plane in which they lie and also their point of intersection.

Show that the plane whose vector equation is contains the line whose vector equation is

Find the equation of the plane determined by the intersection of the lines and

Find the vector equation of the plane passing through the points (3, 4, 2) and (7, 0, 6) and perpendicular to the plane 2x – 5y – 15 = 0. Also, show that the plane thus obtained contains the line

If the lines and are perpendicular, find the value of k and hence find the equation of the plane containing these lines.

Find the coordinates of the point where the line intersect the plane x – y + z – 5 = 0. Also, find the angle between the line and the plane.

Find the vector equation of the plane passing through three points with position vectors and Also, find the coordinates of the point of intersection of this plane and the line

Show that the lines and are coplanar.

Show that the lines and are coplanar ?

Find the equation of a plane which passes through the point (3, 2, 0) and contains the line

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

If the line lies in the plane lx + my – z = 9, then find the value of l² + m²?

Find the values of λ for which the lines and are coplanar.

If the lines and are coplanar, find the values of α.

If the straight lines and are coplanar, find the equations of the planes containing them.

Find the shortest distance between the lines and

Find the shortest distance between the lines and 3x – y – 2z + 4 = 0 = 2x + y + z + 1.

Find the image of the point (0, 0, 0) in the plane 3x + 4y – 6z + 1 = 0.

Find the reflection of the point (1, 2, –1) in the plane 3x – 5y + 4z = 5.

Find the coordinates of the foot of the perpendicular drawn from the point (5, 4, 2) to the line Hence or otherwise deduce the length of the perpendicular.

Find the image of the point with position vector in the plane Also, find the position vectors of the foot of the perpendicular and the equation of the perpendicular line through

Find the coordinates of the foot of the perpendicular drawn from the point (1, 1, 2) to the plane 2x – 2y + 4z + 5 = 0. Also, find the length of the perpendicular.

Find the distance of the point (1, -2, 3) from the plane x – y + z = 5 measured along a line parallel to

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

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

Find the distance of the point with position vector from the point of intersection of the line with the plane

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

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

Find the direction cosines of the unit vector perpendicular to the plane + 1 = 0 passing through the origin.

Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2x – 3y + 4z – 6 = 0.

Find the length and the foot of the perpendicular from the point (1, 3/2, 2) to the plane 2x – 2y + 4z + 5 = 0.

Find the position vector of the foot of the perpendicular and the perpendicular distance from the point P with position vector to the plane Also, find the image of P in the plane.

Write the equation of the plane parallel to XOY - plane and passing through the point (2, –3, 5)

Write the equation of the plane parallel to YOZ - plane and passing through (–4, 1, 0).

Write the equation of the plane passing through point (a, 0, 0), (0, b, 0) and (0, 0, c).

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

Write the value of k for which the planes x – 2y + kz = 4 and 2x + 5y – z = 9 are perpendicular.

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

Write the ratio in which the plane 4x + 5y – 3z = 8 divides the line segment joining points (–2, 1, 5) and (3, 3, 2).

Write the distance between the parallel planes 2x – y + 3z = 4 and 2x – y + 3z = 18.

Write the distance of the plane from the origin.

Write the equation of the plane in scalar product form.

Write a vector normal to the plane .

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

Write the equation of the plane containing the lines and .

Write the position vector of the point where the line meets the plane .

Write the value of k for which the line is perpendicular to the normal to the plane .

Write the angle between the line and the plane x + y + 4 = 0.

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

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

Write the vector equation of the line passing through the point (1, –2, –3) and normal to the plane .

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

Find the vector equation of a plane which is at a distance of 5 units from the origin and its normal vector is .

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

Mark the correct alternative in the following:

The plane 2x – (1 + λ)y + 3λz = 0 passes through the intersection of the planes.

Mark the correct alternative in the following:

The acute angle between the planes 2x – y + z = 6 and x + y + 2z = 3 is

Mark the correct alternative in the following:

The equation of the plane through the intersection of the planes x + 2y + 3z = 4 and 2x + y – z = –5 and perpendicular to the plane 5x + 3y + 6z + 8 = 0 is

Mark the correct alternative in the following:

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

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

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The equation of the plane containing the two lines

and is

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The equation of the plane in scalar product from is

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The distance of the line from the plane is

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The equation of the plane through the line x + y + z + 3 = 0 = 2x – y + 3z + 1 and parallel to the line is

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The vector equation of the plane containing the line and the point is

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A plane meets the coordinate axes at A, B, C such that the centroid of ΔABC is the point (a, b, c). If the equation of the plane is then k =

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The distance between the point (3, 4, 5) and the point where the line meets the plane x + y + z = 17, is

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A vector parallel to the line of intersection of planes and is

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If a plane passes through the point (1, 1, 1) and is perpendicular to the line then its perpendicular distance from the origin is

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The equation of the plane parallel to the lines x – 1 = 2y – 5 = 2z and 3x = 4y – 11 = 3z – 4 and passing through the point (2, 3, 3) is

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The distance of the point (–1, –5, –10) from the point of intersection of the line and the plane is

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The equation of the plane through the intersection of the planes ax + by + cz + d = 0 and lx + my + nz + p = 0 and parallel to the line y = 0, z = 0

Mark the correct alternative in the following:

The equation of the plane which cuts equal intercepts of unit length on the coordinate axes is