July 22, 2024
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R3 – Multiplication Links to Rotation and Rotication – Part 2

The R3 multiplication and rotication formulae also demonstrate how multiplication links to rotication. The post on multiplication use of trigonometry gives the formula for multiplication as:

  • Cx = RA*RB(cos(φAx)*cos(φBx) + cos(φAy)*cos(φBz) + cos(φAz)*cos(φBy))
  • Cy = RA*RB(cos(φAx)*cos(φBz) + cos(φAy)*cos(φBy) + cos(φAz)*cos(φBx))
  • Cz = RA*RB(cos(φAx)*cos(φBy) + cos(φAy)*cos(φBx)+ cos(φAz)*cos(φBz))

Part 1 of this post shows how the rotication formula can be rewritten in terms of Opposite Value types as:

  • ^/value: (xa*xo + ya*zo + za*yo
  • i^/ivalue: (ya*yoxa*zo + za*xo)
  • j^/jvalue: (za*zo + xa*yo + ya*zo)

The multiplication calculations below are from the multiplication use of trigonometry post. They correspond with the and rotication formulae calculations as can be seen from the following:

  • Multiplication Formula
    • Cx = 9.48*8.79(0.665v*0.444^ + 0.496i^*0.273jv +0.559j^*0.853iv)
      • = 83.31*(0.295^ + 0.135v + 0.477^)
      • = 24.57^ + 11.28v + 39.75^
      • = 53.04^

  • Rotication Formula
    • ^/v value: =(xa*xo + ya*zo + za*yo
      • = 6.3v*3.9^ + 4.7i^*2.4jv + 5.3j^* 7.5iv
      • = 24.57^+11.28v + 39.75^
      • = 53.04^

It can be seen from this that:

  • xa*xo = RA*RB(cos(φAx)*cos(φBx))
  • ya*zo = RA*RB(cos(φAy)*cos(φBz))
  • za*yo = RA*RB(cos(φAz)*cos(φBy))

It can be similarly shown that:

  • ya*yo = RA*RB(cos(φAy)*cos(φBy)
  • xa*zo= RA*RB(cos(φAx)*cos(φBz)
  • za*xo= RA*RB(cos(φAz)*cos(φBx)
  • za*zo= RA*RB(cos(φAz)*cos(φBz)
  • xa*yo= RA*RB(cos(φAx)*cos(φBy)
  • ya*zo= RA*RB(cos(φAy)*cos(φBz)

Both formulae implement the multiplication table rules.

Movement using Rotation, Rotication and Multiplication

The different types of movement of a point during R3 multiplication, rotation and rotication are linked. Rotation moves a point to any position on a circle around the origin with the same radius. All the possible circles represent a sphere around the origin. Consequently, rotation cannot move a point away from the sphere.

Rotication is a two part operation. Firstly, the operation is the same as rotation and results in a circular movement of a point. Secondly, it takes the result of the rotation and multiplies it by the magnitude of the axis of rotation, resulting in a linear movement of a point. As a result, a point is able to move anywhere on, inside or outside the sphere. When the axis has a magnitude < 1 the movement is inside the sphere, when = 1 it is on the sphere and when > 1 it is outside the sphere.

This two-part post shows the link between R3 multiplication and rotication. Multiplication is the sum of rotications of π^/2 around the x, y, z axes. As a result, some points cannot be reached directly from other points using multiplication.

For example, nothing can be multiplied with ^ to get v, with i^ to get iv or with j^ to get jv as these are π^ rotations. It can be achieved by 2 multiplications such as i^*j^ = 1^, i^*1^ = jv.

Consequently, rotication is the only operation which allows the direct movement of a point to any other location.

Next: Dot Product

Previous: Multiplication Links Part 1

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