October 5, 2024
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R3 – Multiplication Links to Rotation and Rotication – Part 1

Introduction:

This post starts with a description of long multiplication in R3. It then describes the R3 multiplication links to rotation and rotication and how the different operations move a point.

Long Multiplication

Calculate (5^ + 7iv + 2j^)*(4v + 6iv + 4jv) = (20^ + 30iv + 66jv) by long multiplication and using the R3 multiplication table. For example:

4v                      6iv                   4jv   

5^                       7iv                    2j^  

=======================

20v                  20i^                    30jv

28^                  42iv                    28jv

12^                  8iv                    8jv

============================

20^                  30iv                  66jv

Rotication is a circular movement followed by a linear movement. In contrast, multiplication is the sum of a series of individual rotications of π^/2 around about axes based on each Opposite Value Type and Sign. Therefore there is a direct multiplication link to rotication. As rotication is linked to rotation, there is also a multiplication link to rotation in R3.

(5^ + 7iv + 2j^)*(4v + 6iv + 4jv) = (20^ + 30iv + 66jv)

This is the sum of the following 3 rotications that, as can be seen, match the lines in the long multiplication:

  • π^/2 R↺5^ (4v + 6iv + 4jv) = (20v + 20i^ + 30jv)
  • π^/2 R↺7iv (4v + 6iv + 4jv) = (28^ + 42iv + 28jv)
  • π^/2 R↺2j^ (4v + 6iv + 4jv) = (12^ + 8iv + 8jv)
    • Total (20^ + 30iv + 66jv)

Formula for Rotication

R3 rotication is calculated using the magnitude of the axis, \sqrt{(|x_a|^2 + |y_a|^2 + |z_a|^2)} and the Wave Number formula as follows:

    \[v_{Rotication} = \sqrt{(|x_a|^2 + |y_a|^2 + |z_a|^2)} * ((u_r \cdot v)u_r + sin\theta(u_r \times v) +cos\theta((u_r \times v)\times u_r))\]

Take, for example, (5^ + 7iv + 2j^)*(4v + 6iv + 4jv) = (20^ + 30iv + 66v). The Operator, (5^ + 7iv + 2j^), is the axis of rotation.

First Rotication

Multiplication can be calculated as the sum of the following R3 rotications around the x, y and z-axes where the axes of rotation are (5^, 0, 0), (0, 7iv, 0) and (0, 0, 2j^) and the angle of rotation is π^/2.

x-axis – (5^, 0, 0) – u_r = (^, 0, 0), v = (4v + 6iv + 4jv) , magnitude = √(52) = 5

  • Rotication = 5* ((^, 0, 0).(4v + 6iv + 4jv))*u_r   + (sinπ^/2 (u_r x v)) + (cosπ^/2 )(u_r x v) x u_r)
  • = 5* ((4)*(^, 0, 0)   + (1* (u_r x v)) + 0*(u_r x v) x u_r))
  • = 5* (4v   + ((^, 0, 0) x (4v + 6iv + 4jv))
  • = 5* (4v   + (0*4jv + 0*6iv + 0*4v + ^*4jv + ^*6iv + 0*4v)
  • = 5* (4v   + 4i^ + 6jv )
  • = (20v   + 20i^ + 30jv )

Second Rotication

y-axis – (0, 7iv, 0) – u_r = (0, iv, 0), v = (4v + 6iv + 4jv) ,magnitude = √(72) = 7

  • Rotication = 7* ((0, iv, 0).(4v + 6iv + 4jv))*u_r + (sinπ^/2 (u_r x v)) + (cosπ^/2 )(u_r x v) x u_r)
  • = 7* ((6)*(0, iv, 0)   + (1* (u_r x v)) + 0*(u_r x v) x u_r))
  • = 7* (6iv   + ((0, iv, 0) x (4v + 6iv + 4jv))
  • = 7* (6iv   + (iv*4jv + 0*6iv + 0*4v + 0*4jv + 0*6iv + iv*4v)
  • = 7* (6iv   + 4^ + 4jv)
  • = (28^ + 42iv  + 28jv)

Third Rotication

z-axis – (0, 0, 2j^) u_r= (0, 0, j^), v = (4v + 6iv + 4jv), magnitude = √(22) = 2

  • Rotication = 2* ((0, 0, j^).(4v + 6iv + 4jv))*u_r   + (sinπ^/2 (u_r x v)) + (cosπ^/2 )(u_r x v) x u_r)
  • = 2* ((4)*(0, 0, j^)   + (1* (u_r x v)) + 0*(u_r x v) x u_r))
  • = 2* (4jv   + ((0, 0, j^) x (4v + 6iv + 4jv))
  • = 2* (4jv   + (0*4jv + j^*6iv + j^*4v + 0*4jv + 0*6iv + 0*4v)
  • = 2* (4jv   + 6^ + 4iv)
  • = (12^ + 8iv +8jv)

Total

  • (20v + 20i^ + 30jv ) +(28^ + 42iv + 28jv) + (12^ + 8iv +8jv)
  • = (20^ + 30iv + 66jv )

So, it can be seen that this matches the result of long multiplication above.

Simplifying Rotication Formula

The units of rotation are each made up of a pair of zeros and a unitary and the angle is always π^/2, therefore the formula for multiplication can be simplified. In the simplified formula below, the symbol a represents the operator or axis, o represents the operand, and x, y, z represent the coordinates.

xa*xo + xa*zo + xa*yo + ya*yo+ ya*xo + ya*zo + za*zo + za*yo + za*xo which in this example gives:

  • 5^*4v + 5^*4jv + 5^*6iv + 7iv*6iv + 7iv*4v + 7iv*4jv + 2j^*4jv + 2j^*6iv + 2j^*4v
  • = 20v + 20i^ + 30jv + 42iv + 28jv + 28^ + 8jv + 12^ + 8iv
  • = 20^ + 30iv + 66jv

This matches the result of the long multiplication and the sum of the R3 rotications above.

The formula can be rewritten in terms of Opposite Value types as:

  • ^/v value: (xa*xo + ya*zo + za*yo
  • i^/iv value: (ya*yo+ xa*zo + za*xo)
  • j^/jv value: (za*zo + xa*yo + ya*xo)

(6.3v + 4.7i^ + 5.3j^)*(3.9^ + 7.5iv + 2.4jv) =  (53.04^ + 29.7iv + 16.2j^

This is the sum of the following 3 rotications:

  • π^/2 R↺6.3v (3.9^ + 7.5iv + 2.4jv)
    • = (24.57^ + 15.12iv + 47.25j^)
  • π^/2 R↺4.7i^ (3.9^ + 7.5iv + 2.4jv)
    • = (11.28v + 35.25iv + 18.33jv)
  • π^/2 R↺5.3j^ (3.9^ + 7.5iv + 2.4jv)
    • = ( 39.75^ + 20.67i^ + 12.72jv)
      • Total (53.04^ + 29.7iv + 16.2j^)

The Operator, (6.3v + 4.7i^ + 5.3j^), is the axis of rotation.

Calculating Sum of R3 Rotications

Multiplication can be calculated as the sum of the following R3 rotications around the x, y and z-axes. Calculations are based on the rotication formula above where the axes of rotation are (6.3v, 0, 0), (0, 4.7i^, 0) and (0, 0, 2j^) and the angle of rotation is π^/2.

First Rotication

x-axis – (6.3v, 0, 0) – u_r = (v, 0, 0), v = (3.9^ + 7.5iv + 2.4jv), magnitude = √(6.32) = 6.3

  • Rotication = 6.3* ((v, 0, 0).(3.9^ + 7.5iv + 2.4jv) * u_r  + (sinπ^/2 (u_r x v)) + (cosπ^/2 )(u_r x v) x ur)
  • = 6.3* ((3.9)*(v, 0, 0)   + (1* (u_r x v)) + 0*(u_r x v) x ur))
  • = 6.3* (3.9^   + ((v, 0, 0) x (3.9^ + 7.5iv + 2.4jv) )
  • = 6.3* (3.9^   + (0*2.4jv + 0*7.5iv + 0*3.9^v*2.4jv + v*7.5iv + 0*3.9^)
  • = 6.3* (3.9^   + 2.4iv + 7.5j^ )
  • = (24.57^   + 15.12iv + 47.25j^ )

Second Rotication

y-axis – (0, 4.7i^, 0) – u_r = (0, i^, 0), v = (3.9^ + 7.5iv + 2.4jv), magnitude = √(4.72) = 4.7

  • Rotication = 4.7* ((0, i^, 0).(3.9^ + 7.5iv + 2.4jv) * u_r    + (sinπ^/2 (u_r x v )) + (cosπ^/2 )(u_r x v ) x ur)
  • = 4.7* ((7.5)*(0, i^, 0)   + (1* (u_r x v )) + 0*(u_r x v ) x u_r ))
  • = 4.7* (7.5iv   + ((0, i^, 0) x (3.9^ + 7.5iv + 2.4jv) )
  • = 4.7* (7.5iv   + (i^*2.4jv + 0*7.5iv + 0*3.9^ + 0*2.4jv + 0*7.5iv + i^*3.9^)
  • = 4.7* (7.5iv   + 2.4v + 3.9jv)
  • = (11.28v + 35.25iv + 18.33jv)

Third Rotication

z-axis – (0, 0, 5.3j^) u_r = (0, 0, j^), v = (3.9^ + 7.5iv + 2.4jv), magnitude = √(5.32) = 5.3

  • Rotication = 5.3* ((0, 0, j^).(3.9^ + 7.5iv + 2.4jv) )* u_r    + (sinπ^/2 (u_r x v )) + (cosπ^/2 )(u_r x v ) x ur)
  • = 5.3* ((2.4)*(0, 0, j^)   + (1* (u_r x v )) + 0*(u_r x v ) x u_r ))
  • =5.3* (2.4jv   + ((0, 0, j^) x (3.9^ + 7.5iv + 2.4jv) )
  • = 5.3* (2.4jv   + (0*2.4jv + j^*7.5iv + j^*3.9^ + 0*2.4jv + 0*7.5iv + 0*3.9^)
  • = 5.3* (2.4jv   + 7.5^ + 3.9i^)
  • = (39.75^ + 20.67i^ + 12.72jv)

Total

  • (24.57^   + 15.12iv + 47.25j^ ) + (11.28v + 35.25iv + 18.33jv) + (39.75^ + 20.67i^ + 12.72jv)
  • = (53.04^   + 29.7iv + 16.2j^)

Using the Simplified Rotication Formula

Given (xa, ya, za) are the coordinates of the axis and (xo , yo, zo) are the coordinates of the operand, then the multiplication of two points can be expressed as:

xa*xo + xa*zo + xa*yo + ya*yo+ ya*xo + ya*zo + za*zo + za*yo +za*xo which in this example gives:

  • = 6.3v*3.9^ + 6.3v*2.4jv + 6.3v* 7.5iv + 4.7i^*7.5iv + 4.7i^*3.9^ + 4.7i^*2.4jv + 5.3j^*2.4jv + 5.3j^* 7.5iv + 5.3j^*3.9^
  • = 24.57^ + 15.12iv + 47.25j^ + 35.25iv + 18.33jv +11.28v +12.72jv + 39.75^ + 20.67i^
  • = (53.04^   + 29.7iv + 16.2j^)

This matches the sum of the R3 rotications above.

Conclusion

Part 2 of this post discusses the links between the R3 multiplication and rotication formulae. It also reviews the different types of movement of a point caused by rotation, rotication and multiplication.

Next: Multiplication Links Part 2

Previous: Trigonometry

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