Class 12 Maths - TELANGANA
Three Dimensional Geometry
The Chapter Three Dimensional Geometry in Class 12 Mathematics for Telangana (TSBSE) students extends your knowledge of 2D coordinate geometry into 3D space. You will study direction cosines and direction ratios of a line, equations of lines and planes in space in various forms, angle between two lines, two planes, or a line and a plane, and the shortest distance between two skew lines. This chapter is vital for scoring high in board exams, as it features direct formula-based questions as well as challenging 7-mark long-answer questions that require visualizing geometric figures in three dimensions.
Start Learning FreeKey Concepts
Direction Cosines and Direction Ratios
Direction cosines (l, m, n) are cosines of angles made by a line with the positive x, y, and z axes, satisfying l squared plus m squared plus n squared equals 1. Direction ratios (a, b, c) are numbers proportional to direction cosines.
Equation of a Line in Space
A line in 3D can be defined using a point through which it passes and a parallel vector (vector form), or using a point and direction ratios (Cartesian form).
Equation of a Plane
A plane can be represented in various forms including normal form, passing through a given point and perpendicular to a given vector, or passing through three non-collinear points.
Angle between Lines and Planes
The angle between two lines or two planes is determined using the dot product of their direction vectors or normal vectors respectively.
Shortest Distance between Two Skew Lines
Skew lines are lines in 3D that are neither intersecting nor parallel. The shortest distance between them is measured along the common perpendicular.
Important Formulas
Board Exam Info
In the Telangana (TSBSE) Class 12 Mathematics board examination, Three Dimensional Geometry carries a significant weightage of approximately 8 to 12 marks. Students can regularly expect one 7-mark long-answer question (usually on finding the shortest distance between skew lines or finding the equation of a plane) along with a few 2-mark or 4-mark short questions based on direction cosines and angle between lines.
Frequently Asked Questions
What is the difference between direction cosines and direction ratios?
Direction cosines are unique for a given directed line and their sum of squares equals 1. Direction ratios are any numbers proportional to the direction cosines and are not unique.
How do I prove two lines are skew lines?
To prove two lines are skew, you must show that they are neither parallel nor intersecting. Mathematically, verify that the shortest distance between them is non-zero.
Is vector form or Cartesian form preferred in board exams?
Both forms are equally important in TSBSE exams. You should know how to convert a line or plane equation from vector form to Cartesian form and vice versa, as questions may ask for either.
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