NCERT Solutions for Class 12 Mathematics Chapter 11 Three Dimensional Geometry
NCERT Solutions for Class 12 Mathematics Chapter 11 Three Dimensional Geometry covers direction cosines and direction ratios of a line, the equations of a line and a plane in three-dimensional space, and the angle and distance between lines and planes.
Overview of Topics & Subtopics (Click to Jump)
1. Direction Cosines and Direction Ratios
Short Note:
• Direction cosines (l, m, n) of a line are the cosines of the angles it makes with the x, y, and z axes respectively, and always satisfy l² + m² + n² = 1.
• Direction ratios are any numbers proportional to the direction cosines and are often more convenient to use, since they need not be normalized.
2. Equation of a Line in Space
Short Note:
• The equation of a line passing through a given point with known direction ratios can be expressed in vector form, r = a + λb, or in Cartesian (symmetric) form using the coordinates and direction ratios.
• Two lines in space can be checked for intersection, parallelism, or being skew (neither intersecting nor parallel) by comparing their direction ratios and solving their equations simultaneously.
3. Equation of a Plane
Short Note:
• The general Cartesian equation of a plane is Ax + By + Cz + D = 0, where (A, B, C) represents the direction ratios of the normal to the plane.
• The angle between two planes, the distance of a point from a plane, and the angle between a line and a plane can all be determined using the normal vector of the plane(s) involved.
NCERT Textbook Exercise Questions
Read More & Chapter Resources
These short notes are designed for quick revision. Read the main points once, then use the related material below to practise the same chapter. Revising a concept and checking it with questions can make your preparation more focused.