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Vector Triple Cross Product
Vector Triple Dot Product
Scalar Triple Product
Orthogonal & Orthonormal Vector
Cos Angle of Vectors
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Resources on Linear Algebra

Orthogonal Vector & Orthonormal Vector

By Kardi Teknomo, PhD.

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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


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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