In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin. x + y + z = 1

Mathematics Part-II

Book: Mathematics Part-II

Chapter: 11. Three Dimensional Geometry

Subject: Maths - Class 12^th

Q. No. 4 of Exercise 11.3

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4

In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
x + y + z = 1

Let the coordinate of the foot of ⊥ P from the origin to the given plane be P(x,y,z).

0x – 5y + 0z = 8

Direction ratio (0,-5,0)

= 5

This is the form of

lx + my + nz = d (∴ d = Distance of the normal from the origin.)

Coordinate of the foot (ld,md,nd)

4

Chapter Exercises

Exercise 11.1

Exercise 11.2

Exercise 11.3

Miscellaneous Exercise

1

In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.
z = 2

1

In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.
x + y + z = 1

1

In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.
2x + 3y – z = 5

1

In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.
5y + 8 = 0

2

Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector

3

Find the Cartesian equation of the following planes:

3

Find the Cartesian equation of the following planes:

3

Find the Cartesian equation of the following planes:

4

In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
2x + 3y + 4z – 12 = 0

4

In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
2x + 3y + 4z – 12 = 0

4

In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
3y + 4z – 6 = 0

4

In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
x + y + z = 1

5

Find the vector and cartesian equations of the planes
that passes through the point (1, 0, –2) and the normal to the plane is

5

Find the vector and cartesian equations of the planes
that passes through the point (1,4, 6) and the normal vector to the plane is

6

Find the equations of the planes that passes through three points.
(1, 1, –1), (6, 4, –5), (–4, –2, 3)

6

Find the equations of the planes that passes through three points.
(1, 1, 0), (1, 2, 1), (–2, 2, –1)

7

Find the intercepts cut off by the plane 2x + y – z = 5.

8

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

9

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

10

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

11

Find the equation of the plane 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.

12

Find the angle between the planes whose vector equations are

13

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.
7x + 5y + 6z + 30 = 0 and 3x – y – 10z + 4 = 0

13

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.
2x + y + 3z – 2 = 0 and x – 2y + 5 = 0

13

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.
2x – 2y + 4z + 5 = 0 and 3x – 3y + 6z – 1 = 0

13

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.
2x – 2y + 4z + 5 = 0 and 3x – 3y + 6z – 1 = 0

13

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.
4x + 8y + z – 8 = 0 and y + z – 4 = 0

14

In the following cases, find the distance of each of the given points from the corresponding given plane.
Point Plane

(0, 0, 0) 3x – 4y + 12 z = 3

14

In the following cases, find the distance of each of the given points from the corresponding given plane.
Point Plane

(3, – 2, 1) 2x – y + 2z + 3 = 0

14

In the following cases, find the distance of each of the given points from the corresponding given plane.
Point Plane

(2, 3, – 5) x + 2y – 2z = 9

14

In the following cases, find the distance of each of the given points from the corresponding given plane.
Point Plane

(–6, 0, 0) 2x – 3y + 6z – 2 = 0