INTRODUCTION TO THREE DIMENSIONAL GEOMETRY

Theory

In three-dimensional coordinate geometry, any point in space is represented as \((x, y, z)\), where:

• \(x\) represents distance along x-axis
• \(y\) represents distance along y-axis
• \(z\) represents distance along z-axis

The coordinate axes are defined as:

• x-axis: \(y = 0, z = 0\)
• y-axis: \(x = 0, z = 0\)
• z-axis: \(x = 0, y = 0\)

Solution Roadmap

• Identify definition of x-axis in 3D
• Apply coordinate conditions for x-axis
• Conclude values of y and z

Solution

A point lying on the x-axis satisfies the defining condition of the x-axis in three-dimensional space.

For any point on the x-axis:

\[ y = 0 \quad \text{and} \quad z = 0 \]

while \(x\) can be any real number.

Therefore, the general coordinates of a point on the x-axis are:

\[ (x, 0, 0) \]

Hence, the y-coordinate is \(0\) and the z-coordinate is also \(0\).

Exam Significance

• Very frequently asked in NCERT-based board exams as a direct conceptual question
• Forms the base for understanding direction ratios and lines in 3D geometry
• Important for JEE/competitive exams in identifying axis-based constraints
• Helps in quick elimination in MCQs involving coordinate conditions

Theory

In three-dimensional coordinate geometry, planes are defined by setting one coordinate equal to zero.

• XY-plane: \(z = 0\)
• YZ-plane: \(x = 0\)
• XZ-plane: \(y = 0\)

The XZ-plane is formed by all points that have no displacement along the y-axis.

Solution Roadmap

• Identify the defining condition of the XZ-plane
• Relate it to coordinate form \((x, y, z)\)
• Conclude the value of y-coordinate

Solution

A point lying in the XZ-plane satisfies the defining condition of the XZ-plane in three-dimensional space.

\[ y = 0 \]

This means that the point has no displacement in the y-direction, while \(x\) and \(z\) can take any real values.

Therefore, the general coordinates of such a point are:

\[ (x, 0, z) \]

Hence, the y-coordinate of a point lying in the XZ-plane is always zero.

Exam Significance

• Direct conceptual question frequently asked in CBSE board exams
• Forms the base for understanding planes and equations of planes in 3D
• Important in JEE for identifying constraints in coordinate geometry problems
• Helps in solving MCQs involving location of points relative to coordinate planes

Theory

In three-dimensional coordinate geometry, space is divided into eight parts called octants by the three coordinate planes.

The octant depends on the signs of \((x, y, z)\):

• First octant: \( (+, +, +) \)
• Second octant: \( (-, +, +) \)
• Third octant: \( (-, -, +) \)
• Fourth octant: \( (+, -, +) \)
• Fifth octant: \( (+, +, -) \)
• Sixth octant: \( (-, +, -) \)
• Seventh octant: \( (-, -, -) \)
• Eighth octant: \( (+, -, -) \)

Solution Roadmap

• Check signs of \(x, y, z\) for each point
• Match sign pattern with standard octant definitions
• Assign correct octant

Solution

Exam Significance

• Very important for board exams as direct or table-based questions
• Frequently tested in MCQs in JEE and other entrance exams
• Foundation for 3D visualization and vector geometry
• Helps in quick sign-based elimination techniques in competitive exams

Theory

In three-dimensional geometry, the three coordinate axes form three coordinate planes:

• XY-plane: formed by x-axis and y-axis
• YZ-plane: formed by y-axis and z-axis
• XZ-plane: formed by x-axis and z-axis

Each coordinate plane is obtained by setting one coordinate equal to zero. These planes divide space into eight regions called octants.

Solution Roadmap

• Identify which axes form the XY-plane
• Use plane definition to determine coordinate form
• Recall total number of octants formed

Solution

(i) The x-axis and y-axis together determine the XY-plane.

(ii) Any point lying in the XY-plane has zero displacement along the z-axis. Hence, its coordinates are:

\[ (x, y, 0) \]

(iii) The three coordinate planes divide the space into eight octants.

Exam Significance

• Very common fill-in-the-blank question in board exams
• Tests basic understanding of coordinate planes
• Essential for solving 3D geometry MCQs in competitive exams
• Forms foundation for planes, vectors, and spatial reasoning