Vectors in R2

Definition of Vectors in the Plane

Vectors in the plane (denoted R2) are mathematical objects with both magnitude (length) and direction. They can be represented geometrically as directed line segments or algebraically as ordered pairs (x, y).

• Position Vector: A vector with its initial point at the origin, ending at (x, y).

• Components: The numbers x and y in v = (x, y) are called the components of the vector.

• Equality: Two vectors are equal if they have the same magnitude and direction, regardless of their position in the plane.

Example: The vectors from (0,0) to (−2,1), from (1,2) to (−1,3), and from (3,−1) to (1,0) are all equal if their component differences are (−2,1).

Vector Operations in R2

• Addition:

• Scalar Multiplication: for

• Magnitude (Norm):

• Zero Vector: (0, 0), has zero length and no direction.

• Unit Vector: A vector of length 1. Any nonzero vector v can be converted to a unit vector in its direction by .

• Standard Basis: and ; any vector can be written as .

Example: For and , and .

Dot Product and Its Properties

• Definition: For and , .

• Properties:

  • Commutative:

  • Distributive:

  • Scalar:

  • Norm:

• Geometric Interpretation: , where is the angle between u and v.

• Orthogonality: u and v are perpendicular if and only if .

Example: For and , .

Complex Numbers

Definition and Algebraic Form

• Complex Number: , where and .

• Real Part:

• Imaginary Part:

• Equality: if and only if and .

Arithmetic of Complex Numbers

• Addition:

• Subtraction:

• Multiplication:

• Conjugate:

• Modulus:

• Division:

Example:

Polar and Euler Forms

• Polar Form: , where ,

• Euler's Form:

• Argument: (adjusted for quadrant)

Example:

De Moivre's Theorem and Roots

• De Moivre's Theorem:

• n-th Roots: The n distinct roots of are for

Example: The cube roots of are , , and .

Systems of Linear Equations and Matrices

Linear Systems and Matrix Representation

• Linear Equation:

• System: Multiple linear equations in several variables.

• Coefficient Matrix: Matrix of coefficients of the variables.

• Augmented Matrix: Coefficient matrix with an extra column for the constants.

• Matrix Equation: , where is the coefficient matrix, the variable vector, the constants.

Elementary Row Operations

• Multiply a row by a nonzero constant

• Interchange two rows

• Add a multiple of one row to another

Gaussian and Gauss-Jordan Elimination

• Gaussian Elimination: Reduces a matrix to row-echelon form (upper triangular form).

• Gauss-Jordan Elimination: Reduces a matrix to reduced row-echelon form (each leading 1 is the only nonzero entry in its column).

• Solution Types:

  • Unique solution: System is consistent and has as many leading 1's as variables.

  • Infinitely many solutions: System is consistent but has fewer leading 1's than variables.

  • No solution: System is inconsistent (row of zeros with a nonzero constant in augmented matrix).

Example: The system can be solved by row operations to yield .

Matrices

Matrix Basics

• Matrix: A rectangular array of numbers with size (m rows, n columns).

• Square Matrix: Same number of rows and columns ().

• Main Diagonal: Entries in a square matrix.

• Zero Matrix: All entries are zero.

• Identity Matrix: has 1's on the main diagonal, 0's elsewhere.

Matrix Operations

• Addition/Subtraction: Only defined for matrices of the same size; add/subtract corresponding entries.

• Scalar Multiplication: Multiply every entry by the scalar.

• Matrix Multiplication: is defined; entry is the dot product of row of and column of .

• Transpose: flips rows and columns.

• Inverse: exists for square matrices if .

Example: For , if the determinant is nonzero.

Determinants

• 2x2:

• 3x3: Use cofactor expansion or Sarrus' Rule.

• Properties:

  • Determinant changes sign if two rows are swapped.

  • Determinant is zero if two rows are identical or proportional.

  • Determinant of a triangular matrix is the product of the diagonal entries.

Adjugate and Inverse via Determinant

• Adjugate (Adjoint): The transpose of the cofactor matrix.

• Inverse Formula: if .

Solving Systems: Matrix Inversion and Cramer's Rule

• Matrix Inversion: If and is invertible, .

• Cramer's Rule: For , , where is with column replaced by .

Vectors in R3

Vector Algebra in R3

• Vector:

• Magnitude:

• Unit Vector:

• Standard Basis: , ,

Dot Product in R3

• Definition:

• Angle:

• Orthogonality: iff a and b are perpendicular

Cross Product

• Definition:

• Determinant Form:

• Properties:

  • Anticommutative:

  • Distributive:

  • Parallel Vectors: iff a and b are parallel

  • Magnitude: (area of parallelogram spanned by a and b)

Example: , ,

Equations of Lines and Planes in R3

• Line (Vector Equation): ,

• Plane (Point-Normal Form): or

• Standard Plane Equation:

Systems of Equations in R3: Geometric Interpretation

• Three planes can intersect in a point (unique solution), a line (infinitely many solutions), or have no common intersection (no solution).

• Relative positions depend on the normals and constants in the equations.

HTML Table: Properties of Matrix Operations

Summary Table: Types of Solutions for Linear Systems

Additional info: This guide covers the core topics of vectors, complex numbers, matrices, determinants, and systems of equations as relevant to College Algebra. For more advanced topics (e.g., vector projections, cross product applications, or geometric interpretations of systems), see the worked examples and tutorials referenced in the original material.