Class 12 Maths - BIHAR
Three Dimensional Geometry
The chapter Three Dimensional Geometry in Class 12 Mathematics for Bihar Board (BSEB) students builds upon the vector algebra learned earlier, extending coordinate geometry from two to three dimensions. Students will study the direction cosines and direction ratios of a line, equations of lines and planes in space under various conditions, the angle between two lines, two planes, or a line and a plane, and the shortest distance between two skew lines. This chapter is highly scoring and vital for board exams, frequently featuring direct formula-based questions as well as analytical problems that test spatial visualization.
Start Learning FreeKey Concepts
Direction Cosines and Ratios
Direction cosines (l, m, n) are cosines of angles made by a line with coordinate axes, where l^2 + m^2 + n^2 = 1, and direction ratios are numbers proportional to direction cosines.
Equation of a Line
A line in 3D space can be expressed in vector or Cartesian form, determined either by a passing point and a parallel vector, or by two given points.
Angle Between Two Lines
The angle between two lines is calculated using the dot product of their direction vectors or direction ratios.
Shortest Distance Between Two Lines
For skew lines (lines that are neither parallel nor intersecting), the shortest distance formula uses vector cross products and position vectors.
Equation of a Plane
Planes can be represented in various forms including normal form, passing through a point and perpendicular to a given vector, or passing through three non-collinear points.
Important Formulas
Board Exam Info
In the Bihar Board (BSEB) Class 12 Mathematics examination, Three Dimensional Geometry carries approximately 10 to 14 marks. Questions typically include objective (MCQs), short-answer questions based on finding equations of lines or angles, and long-answer 5-mark questions involving shortest distance between skew lines or finding the equation of a plane.
Frequently Asked Questions
What is the difference between direction cosines and direction ratios?
Direction cosines are unique unit values representing angles with axes satisfying l^2+m^2+n^2=1, whereas direction ratios are any numbers proportional to direction cosines and are not unique.
How do we find if two lines are perpendicular or parallel?
Two lines with direction ratios (a1,b1,c1) and (a2,b2,c2) are parallel if a1/a2 = b1/b2 = c1/c2, and perpendicular if a1a2 + b1b2 + c1c2 = 0.
Are vector proofs mandatory for derivation questions in BSEB exams?
Yes, derivations for the shortest distance between two lines and the equation of a plane passing through given points are frequently asked in long-answer sections.
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