Chapter 11

Chapter 11: Three-Dimensional Geometry

Chapter Overview

This chapter deals with the concept of three-dimensional geometry, which is an essential part of mathematics. It involves the study of points, lines, and planes in three-dimensional space. The chapter focuses on the properties and relationships between these geometric objects, including distance, mid-point, and section formulae. It also covers the concept of the sphere, cylinder, and cone, which are three-dimensional solids. The chapter provides a solid foundation for understanding more advanced concepts in geometry and mathematics.

Learning Objectives

• Understand the concept of three-dimensional geometry
• Learn about points, lines, and planes in three-dimensional space
• Study the properties and relationships between geometric objects
• Apply distance, mid-point, and section formulae
• Understand the concept of the sphere, cylinder, and cone

Important Concepts

Points in a 3D Plane

In a three-dimensional plane, a point is represented by a set of three coordinates (x, y, z). These coordinates specify the position of the point in the plane. For example, consider a point P(2, 3, 4) in a three-dimensional plane. The coordinates (2, 3, 4) specify the position of point P in the plane. The x-coordinate represents the distance of point P from the y-z plane, the y-coordinate represents the distance of point P from the x-z plane, and the z-coordinate represents the distance of point P from the x-y plane.

Lines in a 3D Plane

A line in a three-dimensional plane is a set of points that extend infinitely in two directions. It can be represented by a set of parametric equations. For example, consider a line L that passes through the points P(2, 3, 4) and Q(5, 6, 7) in a three-dimensional plane. The parametric equations of line L can be written as:

x = 2 + 3t y = 3 + 4t z = 4 + 5t

where t is a parameter.

Planes in a 3D Space

A plane in a three-dimensional space is a flat surface that extends infinitely in all directions. It can be represented by a set of parametric equations. For example, consider a plane P that passes through the points A(2, 3, 4), B(5, 6, 7), and C(8, 9, 10) in a three-dimensional space. The parametric equations of plane P can be written as:

x = 2 + 3s - 4t y = 3 + 4s + 5t z = 4 - 2s + 3t

where s and t are parameters.

Distance Formula

The distance between two points (x1, y1, z1) and (x2, y2, z2) in a three-dimensional plane is given by the formula:

√((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)

For example, consider two points P(2, 3, 4) and Q(5, 6, 7) in a three-dimensional plane. The distance between points P and Q can be calculated using the distance formula as:

√((5 - 2)^2 + (6 - 3)^2 + (7 - 4)^2) = √(9 + 9 + 9) = √27

Mid-Point Formula

The mid-point of two points (x1, y1, z1) and (x2, y2, z2) in a three-dimensional plane is given by the formula:

((x1 + x2)/2, (y1 + y2)/2, (z1 + z2)/2)

For example, consider two points P(2, 3, 4) and Q(5, 6, 7) in a three-dimensional plane. The mid-point of points P and Q can be calculated using the mid-point formula as:

((2 + 5)/2, (3 + 6)/2, (4 + 7)/2) = (3.5, 4.5, 5.5)

Section Formula

The section formula for a point that divides a line segment joining two points (x1, y1, z1) and (x2, y2, z2) in the ratio m:n is given by:

(x, y, z) = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n), (mz2 + nz1)/(m+n))

For example, consider a line segment joining two points P(2, 3, 4) and Q(5, 6, 7) in a three-dimensional plane. The point that divides the line segment in the ratio 2:3 can be calculated using the section formula as:

(x, y, z) = ((2(5) + 3(2))/(2+3), (2(6) + 3(3))/(2+3), (2(7) + 3(4))/(2+3)) = ((10 + 6)/5, (12 + 9)/5, (14 + 12)/5) = (4, 21/5, 26/5)

Sphere

A sphere is a three-dimensional solid that is perfectly round and symmetrical about its center. It is defined as the set of all points that are at a fixed distance from a fixed point called the center. For example, consider a sphere with center (0, 0, 0) and radius 5. The equation of the sphere can be written as:

x^2 + y^2 + z^2 = 25

Cylinder

A cylinder is a three-dimensional solid that is formed by a set of parallel and circular bases connected by a curved lateral surface. For example, consider a cylinder with radius 3 and height 6. The equation of the cylinder can be written as:

(x^2 + y^2)/9 = 1, -6 ≤ z ≤ 6

Cone

A cone is a three-dimensional solid that is formed by a set of circular bases connected by a curved lateral surface. For example, consider a cone with radius 4 and height 8. The equation of the cone can be written as:

(x^2 + y^2)/16 = z/8, 0 ≤ z ≤ 8

Advanced Section: Deep-Dive Case Studies and Real-Life Applications

• Case Study 1: Calculating the Distance between Two Points Consider two points P(2, 3, 4) and Q(5, 6, 7) in a three-dimensional plane. Calculate the distance between points P and Q using the distance formula.
• Case Study 2: Finding the Mid-Point of a Line Segment Consider a line segment joining two points P(2, 3, 4) and Q(5, 6, 7) in a three-dimensional plane. Find the mid-point of the line segment using the mid-point formula.
• Case Study 3: Section Formula Consider a line segment joining two points P(2, 3, 4) and Q(5, 6, 7) in a three-dimensional plane. Find the point that divides the line segment in the ratio 2:3 using the section formula.
• Real-Life Application 1: Architecture The concept of three-dimensional geometry is used in various fields such as architecture, engineering, and computer graphics. Architects use the concept of three-dimensional geometry to design buildings and structures.
• Real-Life Application 2: Engineering The concept of three-dimensional geometry is used in various fields such as architecture, engineering, and computer graphics. Engineers use the concept of three-dimensional geometry to design machines and mechanisms.

Advanced Section: Step-by-Step Problem Solving Strategies & Detailed Proofs

• Problem 1: Calculating the Distance between Two Points Consider two points P(2, 3, 4) and Q(5, 6, 7) in a three-dimensional plane. Calculate the distance between points P and Q using the distance formula. Step 1: Write the coordinates of points P and Q. Step 2: Substitute the coordinates into the distance formula. Step 3: Simplify the expression. Step 4: Calculate the distance.

Advanced Section: Higher-Order Thinking Skills (HOTS) Questions

• Question 1: What is the equation of a sphere with center (0, 0, 0) and radius 5?
• Question 2: What is the equation of a cylinder with radius 3 and height 6?
• Question 3: What is the equation of a cone with radius 4 and height 8?

Advanced Section: Previous Year Questions (PYQs) with solutions

• PYQ 1: Calculate the distance between two points P(2, 3, 4) and Q(5, 6, 7) in a three-dimensional plane. Solution: Use the distance formula to calculate the distance between points P and Q.
• PYQ 2: Find the mid-point of a line segment joining two points P(2, 3, 4) and Q(5, 6, 7) in a three-dimensional plane. Solution: Use the mid-point formula to find the mid-point of the line segment.

NCERT Textbook Questions & Detailed Answers

Exercise 11.1

1. Find the distance between the points A(1, 2, 3) and B(4, 5, 6) in a three-dimensional plane. Solution: Use the distance formula to calculate the distance between points A and B.
2. Find the mid-point of the line segment joining the points A(1, 2, 3) and B(4, 5, 6) in a three-dimensional plane. Solution: Use the mid-point formula to find the mid-point of the line segment.

Exercise 11.2

1. Find the equation of a sphere with center (0, 0, 0) and radius 5. Solution: Use the equation of a sphere to find the equation of the sphere with center (0, 0, 0) and radius 5.
2. Find the equation of a cylinder with radius 3 and height 6. Solution: Use the equation of a cylinder to find the equation of the cylinder with radius 3 and height 6.

Exercise 11.3

1. Find the equation of a cone with radius 4 and height 8. Solution: Use the equation of a cone to find the equation of the cone with radius 4 and height 8.
2. Find the point that divides the line segment joining the points A(1, 2, 3) and B(4, 5, 6) in the ratio 2:3. Solution: Use the section formula to find the point that divides the line segment in the ratio 2:3.