Wedge Product of Vectors calculate/provide/represent the Oriented Areas, Hyper-Areas, Volumes and Hyper-Volumes of the Physical Spaces Bounded by the Vectors in form of
resultant \(K\)-Vectors or \(K\)-Dimensional Skew Symmetric Tensors.
For \(N\)-Dimensional Vectors, Wedge Product can be calculated for 2 to \(N\) Vectors .
The Result of a Wedge Product of 2 Vectors is called a Bi-Vector. Given 2 Vectors \(\vec{A_1}\) and \(\vec{A_2}\), the Wedge Product between these 2 is given as \(\vec{A_1} \wedge \vec{A_2}\).
Any Bi-Vector as whole as well as its Individual Components represent Oriented Areas if the 2 Vectors are 2 or 3-Dimensional Vectors or Oriented Hyper-Areas if the 2 Vectors are Higher Dimensional Vectors.
Similarly, the Result of a Wedge Product of 3 Vectors is called a Tri-Vector. Given 3 Vectors \(\vec{A_1}\), \(\vec{A_2}\) and \(\vec{A_3}\), the Wedge Product between these 3 is given as \(\vec{A_1} \wedge \vec{A_2} \wedge \vec{A_3}\).
Any Tri-Vector as a whole as well as its Individual Components represent Oriented Volumes if the 3 Vectors are 3-Dimensional Vectors or Oriented Hyper-Volumes if the 3 Vectors are Higher Dimensional Vectors.
\(\vdots\)
Similarly, the Result of a Wedge Product of K Number of Vectors (where \(3 < K \leq N\)) is called a K-Vector. Given \(K\) Number of Vectors \(\vec{A_1}\), \(\vec{A_2}\), \(\vec{A_3}\) , \(\cdots\), \(\vec{A_K}\), the Wedge Product between these \(K\) Vectors is given as \(\vec{A_1} \wedge \vec{A_2} \wedge \vec{A_3} \wedge \cdots \wedge \vec{A_K}\).
Any K-Vector as a whole as well as its Individual Components represented Oriented Hyper-Volumes.
Hence, for \(N\)-Dimensional Vectors we can have Bi to N-Vectors as a result of wedging 2, 3, 4, \(\cdots N\) Vectors.
Wedge Product of a Vector with itself gives a NULL Bi-Vector. As a corollary, any K-Vector formed as a result of wedging 2 or more Vectors in which 2 or more Vectors are same is a NULL K-Vector.
Altering the order in which Vectors are Wedged to form a K-Vector may alter the Sign of the Individual Components of K-Vector. Odd Number of alterations flip the sign of the Components. Even Number of alterations keep the sign of Components unchanged.
The following demonstrates calculation of Wedge Product for 3 3-Dimensional Vectors \(\vec{A}\), \(\vec{B}\) and \(\vec{C}\) given in Basis Vectors \(\vec{e_1}\), \(\vec{e_2}\) and \(\vec{e_3}\)
Since \(\vec{e_2}\wedge\vec{e_1}=-\vec{e_1}\wedge\vec{e_2}\), \(\vec{e_3}\wedge\vec{e_1}=-\vec{e_1}\wedge\vec{e_3}\), \(\vec{e_3}\wedge\vec{e_2}=-\vec{e_2}\wedge\vec{e_3}\) and, \(\vec{e_1}\wedge\vec{e_1} = \vec{e_1}\wedge\vec{e_1} = \vec{e_1}\wedge\vec{e_1} = 0\) therefore by eliminating and grouping we get,
If \(N\) number of \(N\)-Dimensional Vectors are given, the maximum number of possible K-Vectors that can be formed as a result of Wedging of \(K\) Number of Vectors (where \(K \leq N\)) are \(C(N,K)\).
The number of Components in each \(K\)-Vector thus formed is also \(C(N,K)\). Since K number of Vectors can be wedged in \(K!\) ways, each component of the K-Vector can be given in \(K!\) number of ways based on the order of wedges of the Basis Vectors.
The value of each component is negative in \(\frac{K!}{2}\) ways and positive in \(\frac{K!}{2}\) ways.
The total number of ways in which all components can be given is \(C(N,K) \times K!\).
The Area/Hyper-Area/Volume/Hyper-Volume represented by a \(K\)-Vector is given by the Length/Magnitude of the \(K\)-Vector, which is calculated just like Length/Magnitude of any other Vector (i.e. by calculating the Square Root of Dot Product of the Vector with itself).
The following example calculates the Wedge Product for 3-Dimensional Vectors \(\vec{A_1}\), \(\vec{A_2}\) and \(\vec{A_3}\) given in Identity Orthonormal Basis \(\hat{e_1}\), \(\hat{e_2}\) and \(\hat{e_3}\) (same as \(\hat{i}\), \(\hat{j}\) and \(\hat{k}\)) as defined below
Wedge Product of \(K\) Number of Vectors each of \(N\)-Dimensions can also be calculated by Subtracting Sum of All Tensor Products of Odd Permutations of the Vectors FROM Sum of All Tensor Products of Even Permutations of the Vectors. The K-Vector thus obtained as a result of Wedge Product is in the form of \(K\)-Dimensional Skew Symmetric Tensor.
The following demonstrates the formula for calculating the Wedge Product in this way for 2 Vectors \(\vec{A_1}\) and \(\vec{A_2}\)
In general for K Number of Vectors \(\vec{A_1}\), \(\vec{A_2}\), \(\vec{A_3}\), \(\cdots\), \(\vec{A_K}\)
\(\vec{A_1} \wedge \vec{A_2} \wedge \vec{A_3} \wedge \cdots \wedge \vec{A_K}=\) Sum of All Tensor Products of Even Permutations of the Vectors \(-\) Sum of All Tensor Products of Odd Permutations of the Vectors
A \(K\)-Dimensional Skew Symmetric Tensor of \(N\)-Dimensional Vectors has \(N^K\) elements. Since a K-Vector has \(C(N,K)\) Components and each Component can be given in \(K!\) ways, Each Component of the K-Vector is present K! number of times in the Tensor (out of which \(\frac{K!}{2}\) are negative and \(\frac{K!}{2}\) are positive) and the Total Number of non-zero values in the Tensor is \(C(N,K) \times K!\), half of which are positive and half are negative.
The following example calculates the Wedge Product using Tensor Products for 3-Dimensional Vectors \(\vec{A_1}\) and \(\vec{A_2}\) given in Identity Orthonormal Basis \(\hat{e_1}\), \(\hat{e_2}\) and \(\hat{e_3}\) (same as \(\hat{i}\), \(\hat{j}\) and \(\hat{k}\)) as defined below