Detailed Notes from Toppers: Three-dimensional Geometry (for JEE)

Coordinate Geometry

(Ref: NCERT Class 11, Chapter 12)

• Distance formula in three dimensions: Finds the distance between two points in 3D space.

• Direction cosines and direction ratios of a line: Describe the orientation and direction of a line in space.

• Angle between two lines in space: Determines the angle formed between two lines in 3D space.

• Skew lines: Lines in space that do not intersect or are parallel to each other.

• Planes in space: Equations of planes and properties of planes in 3D space.

Lines and Planes

(Ref: NCERT Class 12, Chapter 11)

• Equations of straight lines in space: Parametric and vector equations of lines in 3D space.

• Intersection of two lines: Finding the point of intersection between two lines in space.

• Coplanar lines: Determining if three or more lines lie on the same plane.

• Parallel lines and perpendicular lines: Identifying parallel and perpendicular lines in 3D space.

Spheres

(Ref: NCERT Class 11, Chapter 12)

• Equation of a sphere: Defines the surface of a sphere in 3D space.

• Tangent planes to a sphere: Finding planes that touch a sphere at a single point.

• Intersection of a sphere with a line and a plane: Finding points where a line or plane intersects a sphere.

Cones and Cylinders

(Ref: NCERT Class 12, Chapter 11)

• Equations of cones and cylinders: Defines the surfaces of cones and cylinders in 3D space.

• Tangent planes to a cone or a cylinder: Finding planes that touch a cone or cylinder at a single point.

• Intersection of a cone or a cylinder with a plane: Finding points where a plane intersects a cone or cylinder.

Quadric surfaces

(Ref: NCERT Class 12, Chapter 11)

• Ellipsoids, hyperboloids, and paraboloids: Equations and properties of quadric surfaces.

Vectors

(Ref: NCERT Class 12, Chapter 10)

• Dot and cross products: Operations performed on vectors.

• Scalar triple products and vector triple products: Scalar and vector quantities computed from three vectors.

• Applications of vectors in geometry and physics: Use of vectors in areas such as mechanics and electromagnetism.

Linear Transformations

(Ref: NCERT Class 12, Chapter 12)

• Matrices: Rectangular arrays of numbers used to represent linear transformations.

• Determinants: Numerical values associated with square matrices.

• Eigenvalues and eigenvectors: Special values and corresponding vectors associated with linear transformations.

• Applications of linear transformations: Use of matrices in areas like systems of linear equations, solving physics problems, etc.

Coordinate systems

(Ref: NCERT Class 12, Chapter 11)

• Cartesian, cylindrical, and spherical coordinate systems: Common coordinate systems used to locate points in space.

• Conversions between coordinate systems: Methods to convert coordinates from one system to another.

Calculus in Three Dimensions

(Ref: NCERT Class 12, Chapter 13)

• Partial differentiation: Differentiation of functions with respect to multiple variables.

• Gradient, divergence, and curl of a vector field: Vector operations used in vector calculus.

• Applications of calculus in three dimensions: Use of calculus in areas such as optimizing functions and calculating work done by a force.

Theorems and Formulae

(Ref: NCERT Class 11 and 12)

• Euler’s Formula: Relates exponential functions to trigonometric functions.

• Menelaus’ Theorem: Relates the lengths of segments of lines intersecting the sides of a triangle.

• Ceva’s Theorem: Relates the ratios of segments of concurrent lines in a triangle.

• Law of Cosines for Triangles: Relates the lengths of the sides and angles of a triangle.

• Law of Sines for Triangles: Relates the ratios of the lengths of the sides of a triangle to the sines of its angles.