Vector fork multiplication and cross multiplication matrix

In this paper, a three-dimensional vector is presented to illustrate the principle of cross-multiplication of vectors and how the cross-multiplication matrices are obtained

1. Calculation principle of vector fork multiplication

A and B are three-dimensional vectors:

A fork by B is generally defined as:

Or

But this is just a definition of a symbol ah, how to get the value of the substitute ?

The key method is to introduce the unit coordinate vector ,

Here I j K to represent the three-dimensional axis, here is just an example, can be extended to more dimensions, but rather abstract

A, by introducing a unit vector, a vector can be converted into an algebraic form:

B. Define operational rules between unit vectors

c, calculate the fork multiply

2, calculate the cross-multiplication matrix

The cross-multiplication result is written in the form of a vector:

The cross-multiplication matrix is obtained from the transformation form:

Which is called a vector of the cross-multiplication matrix.

3, high-dimensional vector to find the cross-multiplication matrix

The calculation of cross-multiplication and cross-multiplication matrices for three-dimensional and three-dimensional vectors can be calculated by defining the arithmetic rules between the unit vectors.

For high-dimensional vectors, this approach is cumbersome and difficult to understand and error prone.

Here's another way to start with a two-dimensional example:

Suppose that vector A is a two-dimensional vector (only two dimensions are used here to make an example easier to understand)

Here is the introduction of an anti-weigh (anti-symmetric) Matrix H:

By calculation, the result is found to be 0.

By the rule of the fork, the result of a fork multiply a is 0:

By contrast, you can find that AH is the cross-multiplication matrix of a vector , when a is a column vector, the cross-multiplication matrix of a vector.

If A is a three-dimensional vector, then H is:

It can be found that the H is constituted by a matrix of opposing calls.

4. Expansion

For Vector point multiplication, the four-tuple multiplier can be defined by the unit vector i J k ... To derive the arithmetic rules between the

Vector fork multiplication and cross multiplication matrix