Formulas

Basic Vector Operations

• Vector Addition:
• Vector Subtraction:
• Scalar Multiplication:
• Vector Between Two Points:

Vector Norm and Distance

• Norm of a Vector in :
• Distance between Points P and Q:
• Normalization (Unit Vector):
• Parallelogram Law:
• Magnitude of Vector Difference Squared:

Dot Product

• Algebraic Dot Product:
• Geometric Dot Product:
• Dot Product with Scalar Multiplication:
• Distributive Property of Dot Product:
• Dot Product and Norm:
• Angle Between Vectors:
• Orthogonality Condition:

Cross Product and Triple Products

• Cross Product:
• Magnitude of Cross Product (Geometric):
• Cross Product and Dot Product Identity:
• Scalar Triple Product (Volume of Parallelepiped):
• Scalar Triple Product as Determinant:
• Coplanarity Test: Three vectors are coplanar if
• Vector Triple Product Identity:

Projections and Reflections

• Projection of vector onto vector :
• Scalar Projection (Component):
• Vector Decomposition: , where and
• Reflection of vector over a line defined by vector :

Geometric Formulas

• Midpoint of a Segment:
• Midpoint Formula (2D):
• Position Vector of Centroid G:
• Centroid of a Triangle:
• Median Vector:
• Section Formula:
• Direction Cosines:
• Direction Cosines Identity:
• Area of a Triangle in 3D:
• Area of Parallelogram from Determinant:

Linear Independence and Basis

• Linear Independence Condition: has only the trivial solution ().
• Condition for Basis in using Determinant: The vectors form a basis if .
• Vector from Coordinates:
• Change-of-Basis Transformation:
• Change-of-Basis Matrix Composition:
• Inverse Change-of-Basis:
• General Change-of-Basis via Standard Basis:

Matrix Properties

• Matrix Multiplication Element:
• Inverse of a 2x2 Matrix: For ,
• Matrix Inverse by Adjugate Method:
• Determinant of a 2x2 Matrix:
• Determinant of a Product:
• Singular Matrix Condition:
• Orthogonal Matrix Condition: or
• Length Preservation by Orthogonal Matrix:

Matrix Rank

• Rank-Nullity Theorem:
• Rank Sum Inequality:
• Rank Product Inequality:
• Sylvester’s Rank Inequality:

Systems of Linear Equations

• System of Equations:
• Cramer’s Rule:
• Inverse Matrix Solution:
• Condition for Unique Solution or Nonsingular Matrix:
• Condition for Nontrivial Solutions in a Homogeneous System:

Analytical Geometry in 2D

• General Form of a Line (2D): (where is the normal vector)
• Intercept Form of a Line (2D):
• Slope of a Line (2D):
• Slope from Angle:
• Slope from General Form: (for )
• Point-Slope Form (2D):
• Perpendicular Slopes (2D):
• Perpendicular Lines Condition (General Form): (for lines and )
• Parallel Lines Condition (2D):
• Coincident Lines Condition (2D):
• Distance from a Point to a Line (2D):

Analytical Geometry in 3D: Lines and Planes

• Parametric Equations of a Line:
• Symmetric Equations of a Line:
• Direction Vector from Intersecting Planes: (for line as intersection of two planes)
• Point-Normal Form of a Plane:
• Standard Form of a Plane:
• Normal Vector from Cross Product:
• Plane Through Three Points: Normal vector , then use point-normal form

Analytical Geometry in 3D: Distances and Angles

• Angle Between Two Lines: (using direction vectors)
• Angle Between Two Planes:
• Angle Between a Line and a Plane:
• Distance from a Point to a Line (3D):
• Distance from a Point to a Plane:
• Distance Between Skew Lines:

Conic Sections: General and Classification

• General Second-Degree Equation:
• Discriminant (Conic Classification):
  • : ellipse
  • : parabola
  • : hyperbola
• Matrix of Quadratic Form:
• Matrix of Quadratic Equation:

Conic Sections: Standard Forms

• Circle (Standard Form):
• Ellipse (Standard Form, Horizontal Major Axis):
• Ellipse (Standard Form, Vertical Major Axis):
• Ellipse (Focus Relation):
• Ellipse (Eccentricity):
• Hyperbola (Standard Form, Horizontal Transverse Axis):
• Hyperbola (Standard Form, Vertical Transverse Axis):
• Hyperbola (Focus Relation):
• Hyperbola (Eccentricity):
• Hyperbola (Asymptotes, Horizontal):
• Hyperbola (Asymptotes, Vertical):
• Parabola (Standard Form, Vertical Axis):
• Parabola (Standard Form, Horizontal Axis):
• Parabola (Canonical Forms): or
• Directrix (Horizontal Ellipse):
• Directrix (Horizontal Hyperbola):
• Latus Rectum Length (Ellipse & Hyperbola):

Parametric Equations

• Circle (Parametric Form): ,
• Ellipse (Parametric Form): ,
• Pythagorean Identity:

Implicit Differentiation and Tangent Lines

• General Implicit Differentiation: For :
• Perpendicularity Condition:
• Tangent Length from External Point: For circle with center , radius , external point :

Coordinate Transformations and Rotations

• Coordinate Translation: ,
• Rotation Angle (Cotangent):
• Rotation Angle (Cosine):
• Half-Angle Formulas:
  • (sign depends on )
  • (always positive)
• Rotation Matrix:
• Coordinate Rotation (Explicit): ,

Orthogonal Invariants

• Orthogonal Invariants (Ellipse/Hyperbola):
• Center Coordinates: Solve
• Orthogonal Invariants (Parabola):
  • where

Quadric Surfaces in 3D

• Sphere (centered at origin):
• Sphere (general center):
• Ellipsoid:
• Elliptic Paraboloid:
• Hyperbolic Paraboloid:
• Hyperboloid of One Sheet:
• Hyperboloid of Two Sheets:
• Circular Cone:

Linear Transformations

• Standard Matrix Construction:
• Rotation Matrix (2D, counterclockwise by ):
• Scaling Matrix (2D):
• Reflection about x-axis:
• Reflection about y-axis:
• Reflection about :
• Shear Matrix (horizontal):
• Projection onto xy-plane (3D):
• Projection onto x-axis (2D):
• Composition of Transformations: where has matrix and has matrix
• Rank-Nullity Theorem (Linear Transformations):

Polar Coordinates

• Polar to Cartesian Conversion: ,
• Cartesian to Polar Conversion: ,
• Line Through Two Points in Polar:
• Line Through Two Points in Polar (Alternative):
• Perpendicular Line in Polar:

Spherical Coordinates

• Spherical to Cartesian Conversion: , ,
• Cartesian to Spherical Conversion: , ,

Conic Sections: Eccentricity and Focus-Directrix

• Ellipse Eccentricity: where (for )
• Hyperbola Eccentricity: where
• Parabola Eccentricity: (always)
• Ellipse Foci (Horizontal Major Axis): where
• Ellipse Directrices (Horizontal Major Axis):
• Hyperbola Foci (Horizontal Transverse Axis): where
• Hyperbola Directrices (Horizontal Transverse Axis):
• Parabola Focus (Right-Opening): for
• Parabola Directrix (Right-Opening): for

Conic Sections in Polar Form

• General Polar Form (Vertical Directrix):
• General Polar Form (Horizontal Directrix):
• Ellipse Polar Form: (focus at origin, right directrix)
• Hyperbola Polar Form: (focus at origin, right directrix)
• Parabola Polar Form: (focus at origin, directrix )