stokes (version 1.2-1)

vector_cross_product: The Vector cross product

Description

The vector cross product \(\mathbf{u}\times\mathbf{v}\) for \(\mathbf{u},\mathbf{u}\in\mathbb{R}^3\) is defined in elementary school as

$$ \mathbf{u}\times\mathbf{v}=\left(u_2v_3-u_3v_2,u_2v_3-u_3v_2,u_2v_3-u_3v_2\right). $$

Function vcp3() is a convenience wrapper for this. However, the vector cross product may easily be generalized to a product of \(n-1\)-tuples of vectors in \(\mathbb{R}^n\), given by package function vector_cross_product().

Vignette vector_cross_product, supplied with the package, gives an extensive discussion of vector cross products, including formal definitions and verification of identities.

Usage

vector_cross_product(M)
vcp3(u,v)

Value

Returns a vector

Arguments

M

Matrix with one more row than column; columns are interpreted as vectors

u,v

Vectors of length 3, representing vectors in \(\mathbb{R}^3\)

Author

Robin K. S. Hankin

Details

A joint function profile for vector_cross_product() and vcp3() is given with the package at vignette("vector_cross_product").

See Also

cross

Examples

vector_cross_product(matrix(1:6,3,2))


M <- matrix(rnorm(30),6,5)
LHS <- hodge(as.1form(M[,1])^as.1form(M[,2])^as.1form(M[,3])^as.1form(M[,4])^as.1form(M[,5]))
RHS <- as.1form(vector_cross_product(M))
LHS-RHS  # zero to numerical precision

# Alternatively:
hodge(Reduce(`^`,sapply(seq_len(5),function(i){as.1form(M[,i])},simplify=FALSE)))