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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

Scalar Triple Product

By Kardi Teknomo, PhD.

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Scalar triple product is a multiplication of three vectors that produces a scalar Scalar Triple Product

Suppose we have three vectors Scalar Triple Product,  Scalar Triple ProductandScalar Triple Product.

First we compute cross product
Scalar Triple Product

Then, the scalar triple product is
Scalar Triple Product

Geometrically, scalar triple product is equal to the volume of parallelepiped. The based area of the parallelepiped is a parallelogram whose area is equal to the absolute value of the cross productScalar Triple Product.

Scalar Triple Product

The interactive program below helps you to compute Scalar triple product algebraically.  You input 3 vectors of the same dimension. The program output is the scalar 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 scalar triple product are

  • Scalar triple product is invariant under cyclic permutation of the vectors, that isScalar Triple Product
  • Scalar triple product is zero if two of the vectors lie on the same plane or are parallel to the same plane (i.e. coplanar) Scalar Triple Product.
  • Three vectors are linearly dependent (coplanar) if and only if their scalar triple product is zeroScalar Triple Product.
  • Three vectors form a basis if and only if their scalar triple product is not zeroScalar Triple Product. The basis is right handed if the scalar triple product is positiveScalar Triple Product and called left handed if the scalar triple product is negativeScalar Triple Product.

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