July 22, 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

The formula for rotication is:

  • Rotication = √(x2a + y2a + z2a)* ((ur.v)*ur   + (sinθ(ur x v)) + (cosθ)(ur x v) x ur)

The Operator, (5^ + 7iv + 2j^), in the formula (5^ + 7iv + 2j^)*(4v + 6iv + 4jv) = (20^ + 30iv + 66v) 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) – ur = (^, 0, 0), v = (4v + 6iv + 4jv) , magnitude = √(52) = 5

  • Rotication = 5* ((^, 0, 0).(4v + 6iv + 4jv))*ur   + (sinπ^/2 (ur x v)) + (cosπ^/2 )(ur x v) x ur)
  • = 5* ((4)*(^, 0, 0)   + (1* (ur x v)) + 0*(ur x v) x ur))
  • = 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) – ur = (0, iv, 0), v = (4v + 6iv + 4jv) ,magnitude = √(72) = 7

  • Rotication = 7* ((0, iv, 0).(4v + 6iv + 4jv))*ur   + (sinπ^/2 (ur x v)) + (cosπ^/2 )(ur x v) x ur)
  • = 7* ((6)*(0, iv, 0)   + (1* (ur x v)) + 0*(ur x v) x ur))
  • = 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^) ur = (0, 0, j^), v = (4v + 6iv + 4jv), magnitude = √(22) = 2

  • Rotication = 2* ((0, 0, j^).(4v + 6iv + 4jv))*ur   + (sinπ^/2 (ur x v)) + (cosπ^/2 )(ur x v) x ur)
  • = 2* ((4)*(0, 0, j^)   + (1* (ur x v)) + 0*(ur x v) x ur))
  • = 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) – ur = (v, 0, 0), magnitude = √(6.32) = 6.3

  • Rotication = 6.3* ((v, 0, 0).(3.9^ + 7.5iv + 2.4jv) *ur   + (sinπ^/2 (ur x v)) + (cosπ^/2 )(ur x v) x ur)
  • = 6.3* ((3.9)*(v, 0, 0)   + (1* (ur x v)) + 0*(ur 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) – ur = (0, i^, 0), magnitude = √(4.72) = 4.7

  • Rotication = 4.7* ((0, i^, 0).(3.9^ + 7.5iv + 2.4jv) *ur   + (sinπ^/2 (ur x v)) + (cosπ^/2 )(ur x v) x ur)
  • = 4.7* ((7.5)*(0, i^, 0)   + (1* (ur x v)) + 0*(ur x v) x ur))
  • = 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^) ur = (0, 0, j^), magnitude = √(5.32) = 5.3

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