Notes from Toppers
Detailed Notes from Toppers: Threedimensional 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.