Matrix Size & Validation
What is Vector?
Vector Scalar Multiple
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
What is a matrix?
Matrix Diagonal Is Diagonal Matrix?
Matrix Basic Operation
Is Equal Matrix?
Matrix Scalar Multiple
Elementary Row Operations
Finding inverse using RREF (Gauss-Jordan)
Finding Matrix Rank using RREF
Is Singular Matrix?
Matrix Generalized Inverse
Solving System of Linear Equations
Linear combination, Span & Basis Vector
Linearly Dependent & Linearly Independent
Change of basis
Matrix Nullity & Null Space
Matrix Eigen Value & Eigen Vector
Matrix Eigen Value & Eigen Vector for Symmetric Matrix
Similarity Transformation and Matrix Diagonalization
Singular Value Decomposition
Resources on Linear Algebra
Vector cross product is also called vector product because the result of the vector multiplication is a vector. It can only be performed for two vectors of the same size.
Geometrically, when you have two vectors on a plane, the cross product will produce another vector perpendicular to the plane span by the two input vectors.
The direction of the cross product vector is following the direction of the thumb finger in your right hand when the four other fingers indicate the angle from the first vector to the second vector, as shown in the following figures.
Computation of cross product
For 1 or two dimensional vector, the cross product produces zero vectors because they do not make a plane yet. Starting from 3 dimensions, the cross product can be computed algebraically using simple arrangement and a simple rule below
The arrangement to compute vector cross product
Simple rule to compute cross product
After the arrangement above, multiply the elements of the vectors in diagonal direction and then minus with the product of elements in counter diagonal direction
Unit vector represent the first, second and third position of vector elements. Thus, here is the final result
For higher dimension, we use the same rule but programmatically, it is easier if we use a formula
Input: vector a, vector b both have equal length
The interactive program of cross product below shows the cross product of two vectors of the same dimension. The program will also show you the internal computation so that you can check your own manual computation. If you click “Random Example” button, the program will generate random input vectors in the right format.
In the applications, cross product is useful for constructing coordinate system mostly in 3-dimensional space.
Some important properties of vector cross product are
Preferable reference for this tutorial is
Teknomo, Kardi (2011) Linear Algebra tutorial. http:\\people.revoledu.com\kardi\ tutorial\LinearAlgebra\