Class 12 Maths - CBSE
Three Dimensional Geometry
The Three Dimensional Geometry chapter in CBSE Class 12 Mathematics builds upon the vector algebra learned earlier, extending coordinate geometry from two dimensions to three. Students learn to describe the positions of points, lines, and planes in space using coordinates and direction ratios. Key topics include direction cosines, equations of lines and planes in various forms, angle between lines and planes, and the shortest distance between two skew lines. This chapter is vital for board exams because it carries a high weightage and features predictable, scoring, long-answer questions that frequently appear in the final CBSE question paper.
Start Learning FreeKey Concepts
Direction Cosines and Direction Ratios
Direction cosines are the cosines of the angles made by a line with the positive X, Y, and Z axes, while direction ratios are any numbers proportional to these direction cosines.
Equation of a Line
A line in space can be determined if passing through a given point and parallel to a given vector, or passing through two given points, expressed in both vector and Cartesian forms.
Angle Between Two Lines
The angle between two intersecting lines is found using the dot product of their direction vectors or direction ratios in Cartesian form.
Shortest Distance Between Two Skew Lines
Skew lines are neither parallel nor intersecting; the shortest distance between them is measured along the common perpendicular line connecting them.
Equation of a Plane
Planes can be represented in various forms depending on given conditions, such as passing through a point and normal to a vector, passing through three non-collinear points, or in intercept form.
Important Formulas
Board Exam Info
In the CBSE Class 12 Mathematics board exam, Three Dimensional Geometry typically carries around 7 to 10 marks. Questions usually include a 4-mark or 6-mark long-answer question based on finding the shortest distance between skew lines or finding the equation of a plane, along with 1-mark objective questions on direction cosines.
Frequently Asked Questions
How do I convert a Cartesian equation of a line into vector form?
Identify the coordinates of the point the line passes through from the numerators (x1, y1, z1) to form vector 'a', and the denominators (a, b, c) to form the parallel vector 'b'.
What is the difference between coplanar lines and skew lines?
Coplanar lines lie in the exact same plane and either intersect or are parallel. Skew lines are non-coplanar, meaning they are neither parallel nor do they ever intersect.
Are vector proofs mandatory for plane and line questions in CBSE exams?
Not always, but knowing both vector and Cartesian forms is essential because questions often ask for a specific form, or converting between them makes calculation easier.
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