Class 11 Mathematics (041) · Chapter 11 NotesIntroduction to Three Dimensional Geometry
Learn about coordinate axes, coordinate planes, and the distance formula in three-dimensional space. Understand how to find coordinates of points and apply the section…
This chapter introduces you to the fundamental concepts of three-dimensional geometry. You will learn how to represent points in space using coordinates, understand the structure of the 3D coordinate system, and calculate distances between points. This knowledge is essential for visualizing and solving problems in three dimensions, which is crucial in fields like physics, engineering, and computer graphics. By the end of this chapter, you will be able to apply these concepts to various real-world scenarios.
What you'll learn
1Understand the concept of coordinate axes and planes in three-dimensional space
2Learn how to determine the coordinates of a point in three-dimensional space
3Calculate the distance between two points in three-dimensional space
4Apply the section formula to find the coordinates of a point dividing a line segment internally in a given ratio
Chapter at a glance
01Coordinate Axes and Coordinate Planes in Three Dimensional Space
02Coordinates of a Point in Three Dimensional Space
03Distance Between Two Points
04Section Formula
Detailed chapter notes
01
Coordinate Axes and Coordinate Planes in Three Dimensional Space
In three-dimensional space, we use three mutually perpendicular planes to locate points. These planes intersect along lines called the x, y, and z-axes. The three planes are the XY-plane (formed by the x and y-axes), the YZ-plane (formed by the y and z-axes), and the ZX-plane (formed by the z and x-axes). These planes divide the space into eight regions called octants. The origin is the point where all three axes intersect. The distances measured along the x, y, and z-axes are taken as positive or negative depending on their direction from the origin.
Three planes intersect at the origin O, forming the x, y, and z-axes
The XY-plane, YZ-plane, and ZX-plane are the three coordinate planes
The coordinate planes divide the space into eight octants
02
Coordinates of a Point in Three Dimensional Space
To find the coordinates of a point in three-dimensional space, we drop a perpendicular from the point to the XY-plane. From the foot of this perpendicular, we draw another perpendicular to the x-axis. The distance from the origin to the foot of this perpendicular along the x-axis is the x-coordinate. The distance from this foot to the point along the perpendicular to the x-axis is the y-coordinate. The distance from the point to the XY-plane is the z-coordinate. These three coordinates (x, y, z) uniquely determine the position of the point in space.
Coordinates of a point P are (x, y, z)
x, y, and z are the distances from the YZ, ZX, and XY planes, respectively
The origin has coordinates (0, 0, 0)
03
Distance Between Two Points
The distance between two points in three-dimensional space can be calculated using the distance formula. For two points P(x1, y1, z1) and Q(x2, y2, z2), the distance PQ is given by the square root of the sum of the squares of the differences in their coordinates. This formula is derived by applying the Pythagorean theorem in three dimensions. The distance from the origin to any point (x, y, z) is given by the square root of the sum of the squares of the coordinates.
Distance from origin to (x, y, z) is sqrt(x^2 + y^2 + z^2)
04
Section Formula
The section formula allows us to find the coordinates of a point that divides a line segment joining two points in a given ratio. For two points P(x1, y1, z1) and Q(x2, y2, z2), the coordinates of the point R that divides the line segment PQ internally in the ratio m:n are given by the section formula. This formula is useful in various geometric applications and helps in locating points that divide a line segment into specific ratios.
Section formula for internal divisionR = ((m x2 + n x1)/(m + n), (m y2 + n y1)/(m + n), (m z2 + n z1)/(m + n))
The section formula can also be applied to external division
Want the complete chapter resources?Topic notes, quizzes and flashcards for Introduction to Three Dimensional Geometry.
In three-dimensional geometry, how many coordinate planes are used to locate a point in space?
A1
B2
C3
D4
Show answer
Answer: (C) 3
Three coordinate planes exist in 3D geometry: the xy-plane, yz-plane, and xz-plane. These planes intersect at the origin and are perpendicular to each other.
Question 02
In three-dimensional coordinate geometry, a point in space is represented by an ordered triplet (x, y, z). What do x, y, and z represent respectively?
Ax is height, y is width, z is depth
Bx is the perpendicular distance from the yz-plane, y is from the xz-plane, and z is from the xy-plane
Cx, y, z are arbitrary distances in any direction
Dx is longitude, y is latitude, z is altitude
Show answer
Answer: (B) x is the perpendicular distance from the yz-plane, y is from the xz-plane, and z is from the xy-plane
In 3D coordinate geometry, x is the perpendicular distance from the yz-plane, y from the xz-plane, and z from the xy-plane. This is the standard definition in CBSE curriculum.
Question 03
What is the distance between points P(1, 2, 3) and Q(4, 6, 15) in three-dimensional space?
A12
B13
C14
D15
Show answer
Answer: (B) 13
Using the distance formula d = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²], we get d = √[(4-1)² + (6-2)² + (15-3)²] = √[9 + 16 + 144] = √169 = 13.
Question 04
If a point P divides the line segment joining A(1, 2, 3) and B(4, 5, 6) in the ratio 1:2 internally, what are the coordinates of P?
A(2, 3, 4)
B(3, 4, 5)
C(1.5, 2.5, 3.5)
D(2.5, 3.5, 4.5)
Show answer
Answer: (A) (2, 3, 4)
Using the section formula for internal division, P = ((1×4 + 2×1)/(1+2), (1×5 + 2×2)/(1+2), (1×6 + 2×3)/(1+2)) = (2, 3, 4).
Question 05
A point lies on the x-axis in three-dimensional space. What are its y and z coordinates?
Ay = 0, z = 0
By = 1, z = 1
Cy = x, z = x
Dy and z can be any real numbers
Show answer
Answer: (A) y = 0, z = 0
Any point on the x-axis has coordinates of the form (x, 0, 0), meaning both y and z coordinates are always zero on the x-axis.
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What are the three coordinate planes in three-dimensional space?
The three coordinate planes are the XY-plane (formed by the x and y-axes), the YZ-plane (formed by the y and z-axes), and the ZX-plane (formed by the z and x-axes).
How do you find the coordinates of a point in three-dimensional space?
To find the coordinates of a point, drop a perpendicular from the point to the XY-plane. From the foot of this perpendicular, draw another perpendicular to the x-axis. The distance from the origin to the foot of this perpendicular along the x-axis is the x-coordinate. The distance from this foot to the point along the perpendicular to the x-axis is the y-coordinate. The distance from the point to the XY-plane is the z-coordinate.
What is the distance formula in three-dimensional space?
The distance between two points P(x1, y1, z1) and Q(x2, y2, z2) in three-dimensional space is given by the formula PQ = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2).
How do you use the section formula to find a point dividing a line segment in a given ratio?
For two points P(x1, y1, z1) and Q(x2, y2, z2), the coordinates of the point R that divides the line segment PQ internally in the ratio m:n are given by R = ((m x2 + n x1)/(m + n), (m y2 + n y1)/(m + n), (m z2 + n z1)/(m + n)).
What are the coordinates of the origin in three-dimensional space?
The coordinates of the origin in three-dimensional space are (0, 0, 0).