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Matrix Size & Validation
Vector Algebra
What is Vector?
Vector Norm
Unit Vector
Vector Addition
Vector Subtraction
Vector Scalar Multiple
Vector Multiplication
Vector Inner Product
Vector Outer Product
Vector Cross Product
Vector Triple Cross Product
Vector Triple Dot Product
Scalar Triple Product
Orthogonal & Orthonormal Vector
Cos Angle of Vectors
Scalar and Vector Projection
Matrix Algebra
What is a matrix?
Special Matrices
Matrix One
Null Matrix
Matrix Diagonal Is Diagonal Matrix?
Identity Matrix
Matrix Determinant
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Is Equal Matrix?
Matrix Transpose
Matrix Addition
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Matrix Multiplication
Matrix Scalar Multiple
Hadamard Product
Horizontal Concatenation
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Elementary Row Operations
Matrix RREF
Finding inverse using RREF (Gauss-Jordan)
Finding Matrix Rank using RREF
Matrix Inverse
Is Singular Matrix?
Linear Transformation
Matrix Generalized Inverse
Solving System of Linear Equations
Linear combination, Span & Basis Vector
Linearly Dependent & Linearly Independent
Change of basis
Matrix Rank
Matrix Range
Matrix Nullity & Null Space
Eigen System
Matrix Eigen Value & Eigen Vector
Symmetric Matrix
Matrix Eigen Value & Eigen Vector for Symmetric Matrix
Similarity Transformation and Matrix Diagonalization
Matrix Power
Orthogonal Matrix
Spectral Decomposition
Singular Value Decomposition
Resources on Linear Algebra

Orthogonal Vector & Orthonormal Vector

By Kardi Teknomo, PhD.
LinearAlgebra

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Two vectors are perpendicular (or orthogonal) to each other if and only if their inner product is zeroOrthogonal Vector. Vectors that perpendicular to each other are also called orthogonal vectors.

When the two vectors that perpendicular to each other also have unit length (i.e. their norm is one), then these vectors are called orthonormal vectors.

The interactive program below will help you to determine whether your input vectors are orthogonal or not. When you click random example button, the program will give you a lot of examples of both orthogonal vectors and not orthogonal vectors.

vector x vector y

Properties

Some important properties of orthogonal & orthonormal vector are

  • Two unit vectors Orthogonal Vectorand Orthogonal Vectorare perpendicular to each other (orthogonal) if and only ifOrthogonal Vector.
  • In 3-dimensional Euclidean space, there are 3 standard unit vectors that orthogonal to each other with special nameOrthogonal Vector,Orthogonal Vectorand Orthogonal Vector. Figure below show the 3 standard orthogonal unit vectors.
    Orthogonal Vector
  • The dot products of the standard orthogonal unit vector:
    • Dot product of the same standard unit vector is oneOrthogonal Vector
    • Dot product of the orthogonal standard unit vector is zeroOrthogonal Vector
  • The cross product of the standard unit vectors:
    • Cross product of the same standard unit vector is zeroOrthogonal Vector
    • Cross product of the orthogonal standard unit vector form a cycle Orthogonal Vector;Orthogonal Vector;Orthogonal Vector;Orthogonal Vector;Orthogonal Vector;Orthogonal Vector;

See also: dot product, cross product, vector norm

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This tutorial is copyrighted.

Preferable reference for this tutorial is

Teknomo, Kardi (2011) Linear Algebra tutorial. http:\\people.revoledu.com\kardi\ tutorial\LinearAlgebra\

 

 
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