Three Dimensional Geometry Class 12 PDF: Complete Guide & Notes

Introduction to Three Dimensional Geometry for Class 12

Three dimensional geometry is a vital chapter in Class 12 NCERT Mathematics that deals with points, lines, and planes in space. Unlike two dimensional geometry, this branch introduces an additional coordinate axis to represent depth, making it essential for understanding real-world spatial problems.

Key concepts include:

• 3D coordinate system with $x$, $y$, and $z$ axes
• Representation of points as $(x, y, z)$
• Distance between two points in space
• Midpoint and section formula in 3D

This chapter forms the foundation for many applications in physics, engineering, and computer graphics, making it important for CBSE exams and beyond.

Coordinate System and Distance Formula in 3D

In three dimensional geometry, every point is represented by an ordered triplet $(x, y, z)$ corresponding to its position along the $x$, $y$, and $z$ axes.

Distance Formula

The distance $d$ between two points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ is given by:

$$ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} $$

Midpoint Formula

The midpoint $M$ of segment $P_1P_2$ is:

$$ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right) $$

These formulas extend the familiar 2D concepts to 3D and are frequently tested in Class 12 exams.

Direction Ratios and Direction Cosines Explained

Direction ratios (DRs) and direction cosines (DCs) describe the orientation of a line in 3D space.

• Direction Ratios are any three numbers proportional to the direction cosines of a line.
• Direction Cosines are the cosines of the angles that the line makes with the $x$, $y$, and $z$ axes, denoted as $l$, $m$, and $n$.

They satisfy the relation:

$$ l^2 + m^2 + n^2 = 1 $$

If $(a, b, c)$ are direction ratios, then the direction cosines are:

$$ \left(\frac{a}{\sqrt{a^2 + b^2 + c^2}}, \frac{b}{\sqrt{a^2 + b^2 + c^2}}, \frac{c}{\sqrt{a^2 + b^2 + c^2}}\right) $$

Understanding these helps in writing the equation of a line and solving related problems.

Equations of a Line in Three Dimensional Geometry

There are two common forms to represent a line in 3D:

1. Symmetric Form:

$$ \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} $$

Here, $(x_1, y_1, z_1)$ is a point on the line and $(a, b, c)$ are direction ratios.

2. Parametric Form:

$$ x = x_1 + at, \quad y = y_1 + bt, \quad z = z_1 + ct $$

where $t$ is a parameter.

Worked Example

Find the symmetric form of the line passing through $(2, -1, 3)$ with direction ratios $3, -4, 12$.

Solution:

$$ \frac{x - 2}{3} = \frac{y + 1}{-4} = \frac{z - 3}{12} $$

This form is useful for finding intersections and angles between lines.

Understanding the Equation of a Plane in 3D Space

A plane in 3D is defined by the equation:

$$ Ax + By + Cz + D = 0 $$

where $(A, B, C)$ is the normal vector perpendicular to the plane.

Key Points:

• The coefficients $A$, $B$, and $C$ are direction ratios of the normal to the plane.
• The distance $d$ from the origin to the plane is:

$$ d = \frac{|D|}{\sqrt{A^2 + B^2 + C^2}} $$

Worked Example

Find the equation of the plane passing through point $(1, 2, 3)$ with normal vector $(4, -5, 6)$.

Solution: Substitute into the plane equation:

$$ 4(x - 1) - 5(y - 2) + 6(z - 3) = 0 $$

Simplify:

$$ 4x - 4 - 5y + 10 + 6z - 18 = 0 \Rightarrow 4x - 5y + 6z - 12 = 0 $$

This equation can be used to solve problems involving planes in Class 12.

Comparing Lines and Planes: Key Differences

Understanding the differences between lines and planes in 3D helps clarify their properties and equations.

This comparison aids in visualizing and solving geometry problems efficiently.

Tips for Using the Three Dimensional Geometry Class 12 PDF Effectively

To make the most of your three dimensional geometry class 12 pdf:

• Start with Definitions: Understand terms like direction ratios, cosines, and normal vectors.
• Visualize Problems: Draw diagrams to grasp spatial relationships.
• Memorize Key Formulas: Keep formulas for distance, section, and equations handy.
• Practice Examples: Solve NCERT solved problems and exercises regularly.
• Review Mistakes: Analyze errors to avoid repeating them in exams.

Consistent practice using the PDF will build confidence and improve your exam performance.