Cross product and cross product matrix vector

In this paper, three-dimensional vector to illustrate the vector cross product cross product matrix calculation principle and how to strike

1, the vector cross product of calculation principle

a, b are three-dimensional vectors:

$a=({a_1},{a_2},{a_3})$

$b=({b_1},{b_2},{b_3})$

b a cross product is generally defined as:

$a{\times}b$ or $a{\otimes}b$

But this is just the definition of a symbol of, ah, how to get specific algebraic value it

The key is to coordinate the introduction of a unit vector ,

Ijk used here to represent the three-dimensional coordinate axes where only an example, can be extended to more dimensions, but rather abstract

a, by introducing a unit vector, the vector can be converted to the algebraic form:

$a {\ rm {=}} {a_1} i {j} + {a_3 a_2 k}$

${\ Rm {b}} = {{\ rm {b}} _ {1} i + j + {} b_2 B_3} k$

b, is defined between the unit vector arithmetic rule

$i*i=0$ $j * j = 0$ $k*k=0$

$i*j=k$ $j*k=i$ $k*i=j$

$j*i=-k$ $k*j=-i$ $i*k=-j$

c, calculate the cross product

$a{\times}b=({a_1}i+{a_2}j+{a_3}k)*({b_1}i+{b_2}j+{b_3}k)$

$a{\times}b=({a_2}{b_3}-{a_3}{b_2})i+({a_3}{b_1}-{a_1}{b_3})j+({a_1}{b_2}-{a_2}{b_1})k$

2, calculate the cross product matrix

$a{\times}b=({a_2}{b_3}-{a_3}{b_2})i+({a_3}{b_1}-{a_1}{b_3})j+({a_1}{b_2}-{a_2}{b_1})k$

The vector cross product results written in the form of:

$a{\times}b=\left[\begin{array}{l} {a_2}{b_3}-{a_3}{b_2}\\ {a_3}{b_1}-{a_1}{b_3}\\ {a_1}{b_2}-{a_2}{b_1} \end{array}\right]$

Transformation matrix cross product obtained in the form:

$a{\times}b={\left[a\right]_\times}b=\left[{\begin{array}{*{20}{c}} 0&{-{a_3}}&{{a_2}}\\ {{a_3}}&0&{-{a_1}}\\ {-{a_2}}&{{a_1}}&0 \end{array}}\right]\left[{\begin{array}{*{20}{c}} {{b_1}}\\ {b{}_2}\\ {{b_3}} \end{array}}\right]$

Wherein ${\left[a\right]_\times}$ it called a vector cross product matrix.

3, high-dimensional vector cross product matrix to strike

For three-dimensional and three-dimensional fork by the following vector cross product is calculated and the matrix can be calculated by obtaining an arithmetic rule is defined between the unit vector.

For high-dimensional vector, this method is difficult to understand a bit tedious and error-prone.

Here is another method, to give a two-dimensional example:

A two-dimensional vector is assumed that a vector (here, only a two-dimensional example is to allow easy understanding)

$a=\left({\begin{array}{*{20}{c}} {{a_1}}&{{a_2}} \end{array}}\right)$

Here introducing an antisymmetric (anti-symmetric) matrix H:

$H=\left[{\begin{array}{*{20}{c}} 0&{-1}\\ 1&0 \end{array}}\right]$

By calculation $aH{a^T}$ , the result is found 0

The results from the cross product of the rule, a cross product a 0:

$a{\times}a={\left[a\right]_\times}a=0$

By comparison, it was found aH is a vector cross product matrix , when A is a column vector ${a^T}H$ by a vector of a matrix fork.

If a three-dimensional vector, H is then:

$H=\left[{\begin{array}{*{20}{c}} {{H_1}}\\ {{H_2}}\\ {{H_3}} \end{array}}\right]$ ${H_1}=\left[{\begin{array}{*{20}{c}} 0&0&0\\ 0&0&{-1}\\ 0&{-1}&0 \end{array}}\right]$ ${H_2}=\left[{\begin{array}{*{20}{c}} 0&0&1\\ 0&0&0\\ {-1}&0&0 \end{array}}\right]$ ${H_3}=\left[{\begin{array}{*{20}{c}} 0&{-1}&0\\ 1&0&0\\ 0&0&0 \end{array}}\right]$

H is found to be composed of one antisymmetric matrix.

If the number is a dimensional vector p, H that have $\frac{{p(p-1)}}{2}$ sub-matrix.

4, expansion

For dot vector, quaternion multiplication can be derived by calculation rules are defined between the unit vector ijk ....

Cross product and cross product matrix vector

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