HSSLIVE Plus One Maths Chapter 12: Introduction to Three Dimensional Geometry Notes

This chapter extends geometric concepts from the plane to three-dimensional space. Students will learn coordinate systems in space, distance formulas, equations of lines and planes, and relationships between geometric objects in 3D. Through visualization exercises and algebraic techniques, students develop spatial reasoning abilities essential for fields like engineering, computer graphics, physics, and advanced mathematics. This introduction to 3D geometry provides the foundation for multivariable calculus and vector analysis.

Chapter 12: Introduction to Three Dimensional Geometry

Coordinate System:

Three mutually perpendicular axes (x, y, z) intersecting at origin O.

Distance Formula:

Distance between points (x₁, y₁, z₁) and (x₂, y₂, z₂) is: d = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²]

Section Formula:

Point dividing the line joining (x₁, y₁, z₁) and (x₂, y₂, z₂) in ratio m

internally: (x, y, z) = [(mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n), (mz₂+nz₁)/(m+n)]

Direction Cosines:

For a line making angles α, β, γ with positive x, y, z axes respectively:

• Direction cosines: l = cos α, m = cos β, n = cos γ
• l² + m² + n² = 1

Direction Ratios:

• If a, b, c are proportional to direction cosines, they are called direction ratios.
• For any line, if a, b, c are direction ratios, then direction cosines are: a/√(a²+b²+c²), b/√(a²+b²+c²), c/√(a²+b²+c²)