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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
Matrix Sum
Matrix Trace
Matrix Basic Operation
Is Equal Matrix?
Matrix Transpose
Matrix Addition
Matrix Subtraction
Matrix Multiplication
Matrix Scalar Multiple
Hadamard Product
Horizontal Concatenation
Vertical Concatenation
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

Triple Cross Product

By Kardi Teknomo, PhD.

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Triple cross product is also called vector triple product. It is a multiplication of three vectors that produces a vector Triple Cross Product

Suppose we have three vectors Triple Cross Product,  Triple Cross ProductandTriple Cross Product.

First we compute cross product
Triple Cross Product

Then, the triple cross product is
Triple Cross Product

Geometrically, vector triple product is perpendicular to Triple Cross Product and lies in the plane span by Triple Cross Product and Triple Cross Product.
Triple Cross Product


The interactive program below helps you to compute triple cross product algebraically.  You input 3 vectors of the same dimension. The program output is the vector triple product. Random example button will generate random vectors at the right format.

vector a vector b vector c x x


Some important properties of related to triple cross product are

  • Vector triple product is zero when any of the two vectors are collinear. Collinear means they lie on the same line or parallel lines.
  • Vector triple product Triple Cross Productfor non-collinear vectors has unique linear combination expansion in the form of Triple Cross Productwhere Triple Cross ProductandTriple Cross Product . Thus, Triple Cross Product
  • Similarly, Triple Cross Product
  • Summation of triple cross product in a cycle is zeroTriple Cross Product. This property is called Jacobi identity

See also: Triple dot product, cross product, inner product, scalar triple product

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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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