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^{^} + 7i^v + 2j^{^})*(4^v + 6i^v + 4j^v) = (20^{^} + 30i^v + 66j^v) by long multiplication and using the R3 multiplication table. For example:

4^v 6i^v 4j^v

5^{^} 7i^v 2j^{^}

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

20^v 20i^{^} 30j^v

28^{^} 42i^v 28j^v

12^{^} 8i^v 8j^v

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

20^{^} 30i^v 66j^v

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^{^} + 7i^v + 2j^{^})*(4^v + 6i^v + 4j^v) = (20^{^} + 30i^v + 66j^v)

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

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:

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

Calculating Sum of R3 Rotications to Show Link to Multiplication

Take, for example, (5^{^} + 7i^v + 2j^{^})*(4^v + 6i^v + 4j^v) = (20^{^} + 30i^v + 66^v). The Operator, (5^{^} + 7i^v + 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, 7i^v, 0) and (0, 0, 2j^{^}) and the angle of rotation is π^{^}/2.

x-axis – (5^{^}, 0, 0) – $u_r$ = (^{^}, 0, 0), $v$ = (4^v + 6i^v + 4j^v) , magnitude = √(5²) = 5

Rotication = 5* ((^{^}, 0, 0).(4^v + 6i^v + 4j^v))* $u_r$ + (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* (4^v + ((^{^}, 0, 0) x (4^v + 6i^v + 4j^v))
= 5* (4^v + (0*4j^v + 0*6i^v + 0*4^v + ^{^}*4j^v + ^{^}*6i^v + 0*4^v)
= 5* (4^v + 4i^{^} + 6j^v )
= (20^v + 20i^{^} + 30j^v )

Second Rotication

y-axis – (0, 7i^v, 0) – $u_r$ = (0, i^v, 0), $v$ = (4^v + 6i^v + 4j^v) ,magnitude = √(7²) = 7

Rotication = 7* ((0, i^v, 0).(4^v + 6i^v + 4j^v))* $u_r$ + (sinπ^{^}/2 ( $u_r$ x $v$ )) + (cosπ^{^}/2 )( $u_r$ x $v$ ) x $u_r$ )
= 7* ((6)*(0, i^v, 0) + (1* ( $u_r$ x $v$ )) + 0*( $u_r$ x $v$ ) x $u_r$ ))
= 7* (6i^v + ((0, i^v, 0) x (4^v + 6i^v + 4j^v))
= 7* (6i^v + (i^v*4j^v + 0*6i^v + 0*4^v + 0*4j^v + 0*6i^v + i^v*4^v)
= 7* (6i^v + 4^{^} + 4j^v)
= (28^{^} + 42i^v + 28j^v)

Third Rotication

z-axis – (0, 0, 2j^{^}) $u_r$ = (0, 0, j^{^}), $v$ = (4^v + 6i^v + 4j^v), magnitude = √(2²) = 2

Rotication = 2* ((0, 0, j^{^}).(4^v + 6i^v + 4j^v))* $u_r$ + (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* (4j^v + ((0, 0, j^{^}) x (4^v + 6i^v + 4j^v))
= 2* (4j^v + (0*4j^v + j^{^}*6i^v + j^{^}*4^v + 0*4j^v + 0*6i^v + 0*4^v)
= 2* (4j^v + 6^{^} + 4i^v)
= (12^{^} + 8i^v +8j^v)

Total

(20^v + 20i^{^} + 30j^v ) +(28^{^} + 42i^v + 28j^v) + (12^{^} + 8i^v +8j^v)

= (20^{^} + 30i^v + 66j^v )

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.

x_a*x_o + x_a*z_o + x_a*y_o + y_a*y_o+ y_a*x_o + y_a*z_o + z_a*z_o + z_a*y_o + z_a*x_o which in this example gives:

5^{^}*4^v + 5^{^}*4j^v + 5^{^}*6i^v + 7i^v*6i^v + 7i^v*4^v + 7i^v*4j^v + 2j^{^}*4j^v + 2j^{^}*6i^v + 2j^{^}*4^v
= 20^v + 20i^{^} + 30j^v + 42i^v + 28j^v + 28^{^} + 8j^v + 12^{^} + 8i^v
= 20^{^} + 30i^v + 66j^v

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:

^{^}/^vvalue: (x_a*x_o + y_a*z_o + z_a*y_o)
i^{^}/i^vvalue: (y_a*y_o+ x_a*z_o + z_a*x_o)
j^{^}/j^vvalue: (z_a*z_o + x_a*y_o + y_a*x_o)

Example 2 of R3 Multiplication Links

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

This is the sum of the following 3 rotications:

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

The Operator, (6.3^v + 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 (6.3^v, 0, 0), (0, 4.7i^{^}, 0) and (0, 0, 2j^{^}) and the angle of rotation is π^{^}/2.

First Rotication

x-axis – (6.3^v, 0, 0) – $u_r$ = (^v, 0, 0), $v$ = (3.9^{^} + 7.5i^v + 2.4j^v), magnitude = √(6.3²) = 6.3

Rotication = 6.3* ((^v, 0, 0).(3.9^{^} + 7.5i^v + 2.4j^v) * $u_r$ + (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^{^} + 7.5i^v + 2.4j^v) )
= 6.3* (3.9^{^} + (0*2.4j^v + 0*7.5i^v + 0*3.9^{^}+ ^v*2.4j^v + ^v*7.5i^v + 0*3.9^{^})
= 6.3* (3.9^{^} + 2.4i^v + 7.5j^{^} )
= (24.57^{^} + 15.12i^v + 47.25j^{^} )

Second Rotication

y-axis – (0, 4.7i^{^}, 0) – $u_r$ = (0, i^{^}, 0), $v$ = (3.9^{^} + 7.5i^v + 2.4j^v), magnitude = √(4.7²) = 4.7

Rotication = 4.7* ((0, i^{^}, 0).(3.9^{^} + 7.5i^v + 2.4j^v) * $u_r$ + (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* (7.5i^v + ((0, i^{^}, 0) x (3.9^{^} + 7.5i^v + 2.4j^v) )
= 4.7* (7.5i^v + (i^{^}*2.4j^v + 0*7.5i^v + 0*3.9^{^} + 0*2.4j^v + 0*7.5i^v + i^{^}*3.9^{^})
= 4.7* (7.5i^v + 2.4^v + 3.9j^v)
= (11.28^v + 35.25i^v + 18.33j^v)

Third Rotication

z-axis – (0, 0, 5.3j^{^}) $u_r$ = (0, 0, j^{^}), $v$ = (3.9^{^} + 7.5i^v + 2.4j^v), magnitude = √(5.3²) = 5.3

Rotication = 5.3* ((0, 0, j^{^}).(3.9^{^} + 7.5i^v + 2.4j^v) )* $u_r$ + (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* (2.4j^v + ((0, 0, j^{^}) x (3.9^{^} + 7.5i^v + 2.4j^v) )
= 5.3* (2.4j^v + (0*2.4j^v + j^{^}*7.5i^v + j^{^}*3.9^{^} + 0*2.4j^v + 0*7.5i^v + 0*3.9^{^})
= 5.3* (2.4j^v + 7.5^{^} + 3.9i^{^})
= (39.75^{^} + 20.67i^{^} + 12.72j^v)

Total

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

Using the Simplified Rotication Formula

Given (x_a, y_a, z_a) are the coordinates of the axis and (x_o , y_o, z_o) are the coordinates of the operand, then the multiplication of two points can be expressed as:

x_a*x_o + x_a*z_o + x_a*y_o + y_a*y_o+ y_a*x_o + y_a*z_o + z_a*z_o + z_a*y_o +z_a*x_o which in this example gives:

= 6.3^v*3.9^{^} + 6.3^v*2.4j^v + 6.3^v* 7.5i^v + 4.7i^{^}*7.5i^v + 4.7i^{^}*3.9^{^} + 4.7i^{^}*2.4j^v + 5.3j^{^}*2.4j^v + 5.3j^{^}* 7.5i^v + 5.3j^{^}*3.9^{^}
= 24.57^{^} + 15.12i^v + 47.25j^{^} + 35.25i^v + 18.33j^v +11.28^v +12.72j^v + 39.75^{^} + 20.67i^{^}
= (53.04^{^} + 29.7i^v + 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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R3 – Multiplication Links to Rotation and Rotication – Part 1

Introduction:

Long Multiplication

R3 Multiplication Link to Rotation and Rotication

Example 1 of R3 Multiplication Links

Formula for Rotication

Calculating Sum of R3 Rotications to Show Link to Multiplication

First Rotication

Second Rotication

Third Rotication

Simplifying Rotication Formula

Example 2 of R3 Multiplication Links

Calculating Sum of R3 Rotications

First Rotication

Second Rotication

Third Rotication

Using the Simplified Rotication Formula

Conclusion

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