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
R3 Multiplication Link to Rotation and Rotication
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.
Example 1 of R3 Multiplication Links
(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, and the Wave Number formula as follows:
Calculating Sum of R3 Rotications to Show Link to Multiplication
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) – = (^, 0, 0), = (4v + 6iv + 4jv) , magnitude = √(52) = 5
- Rotication = 5* ((^, 0, 0).(4v + 6iv + 4jv))* + (sinπ^/2 ( x )) + (cosπ^/2 )( x ) x )
- = 5* ((–4)*(^, 0, 0) + (1* ( x )) + 0*( x ) x ))
- = 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) – = (0, iv, 0), = (4v + 6iv + 4jv) ,magnitude = √(72) = 7
- Rotication = 7* ((0, iv, 0).(4v + 6iv + 4jv))* + (sinπ^/2 ( x )) + (cosπ^/2 )( x ) x )
- = 7* ((6)*(0, iv, 0) + (1* ( x )) + 0*( x ) x ))
- = 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^) = (0, 0, j^), = (4v + 6iv + 4jv), magnitude = √(22) = 2
- Rotication = 2* ((0, 0, j^).(4v + 6iv + 4jv))* + (sinπ^/2 ( x )) + (cosπ^/2 )( x ) x )
- = 2* ((–4)*(0, 0, j^) + (1* ( x )) + 0*( x ) x ))
- = 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)
Example 2 of R3 Multiplication Links
(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^)
- = ( 39.75^ + 20.67i^ + 12.72jv)
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) – = (v, 0, 0), = (3.9^ + 7.5iv + 2.4jv), magnitude = √(6.32) = 6.3
- Rotication = 6.3* ((v, 0, 0).(3.9^ + 7.5iv + 2.4jv) * + (sinπ^/2 ( x )) + (cosπ^/2 )( x ) x ur)
- = 6.3* ((–3.9)*(v, 0, 0) + (1* ( x )) + 0*( x ) 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) – = (0, i^, 0), = (3.9^ + 7.5iv + 2.4jv), magnitude = √(4.72) = 4.7
- Rotication = 4.7* ((0, i^, 0).(3.9^ + 7.5iv + 2.4jv) * + (sinπ^/2 ( x )) + (cosπ^/2 )( x ) x ur)
- = 4.7* ((–7.5)*(0, i^, 0) + (1* ( x )) + 0*( x ) x ))
- = 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^) = (0, 0, j^), = (3.9^ + 7.5iv + 2.4jv), magnitude = √(5.32) = 5.3
- Rotication = 5.3* ((0, 0, j^).(3.9^ + 7.5iv + 2.4jv) )* + (sinπ^/2 ( x )) + (cosπ^/2 )( x ) x ur)
- = 5.3* ((–2.4)*(0, 0, j^) + (1* ( x )) + 0*( x ) x ))
- =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
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