MATHEMATICS S5 UNIT 8: Points, Straight Lines, Planes and Sphere in 3D.

The course “Points, Straight Lines, Planes and Sphere in 3D” is a fundamental topic in three-dimensional analytical solid geometry. It’s designed to provide a comprehensive understanding of geometric objects in 3D space using coordinate systems and vector approaches.

Key Concepts Covered:

• Points in 3D: Locating points using three coordinates (x, y, z), distance formula between two points, midpoint formula, and centroid of a geometric figure.
• Straight Lines in 3D:
  • Equations of lines (vector, parametric, and Cartesian forms).
  • Direction cosines and direction ratios.
  • Angle between two lines.
  • Condition for coplanarity of two lines.
  • Shortest distance between two skew lines.
  • Intersection of lines.
  • Projection of a segment on a line.
• Planes in 3D:
  • Equations of planes (vector, Cartesian, intercept, and normal forms).
  • Equation of a plane passing through given points.
  • Length of the perpendicular from a point to a plane.
  • Angle between two planes.
  • Angle between a line and a plane.
  • Intersection of lines and planes.
  • Intersection of two or three planes.
  • Image of a point with respect to a plane.
  • Bisectors of angles between two planes.
• Sphere in 3D:
  • Definition and equation of a sphere (standard and general forms).
  • Equation of a sphere passing through four given points.
  • Plane sections of a sphere (circles).
  • Intersection of two spheres.
  • Equation of a circle.
  • Sphere through a given circle.
  • Intersection of a sphere and a line.
  • Tangent plane to a sphere.
  • Radical plane and coaxial systems of spheres.
  • Angle of intersection of two spheres (including orthogonal spheres).