R3 – Multiplication Links to Rotation and Rotication – Part 2

R3 Multiplication Link between Formulae:

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:

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

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(0.665^v*0.444^{^} + 0.496i^{^}*0.273j^v +0.559j^{^}*0.853i^v)
    • = 83.31*(0.295^{^} + 0.135^v + 0.477^{^})
    • = 24.57^{^} + 11.28^v + 39.75^{^}
    • = 53.04^{^}

• Rotication Formula
  • ^{^}/^v value: =(x_a*x_o + y_a*z_o + z_a*y_o) 
    • = 6.3^v*3.9^{^} + 4.7i^{^}*2.4j^v + 5.3j^{^}* 7.5i^v
    • = 24.57^{^}+11.28^v + 39.75^{^}
    • = 53.04^{^}

It can be seen from this that:

• x_a*x_o = RA*RB(cos(φAx)*cos(φBx))
• y_a*z_o = RA*RB(cos(φAy)*cos(φBz))
• z_a*y_o = RA*RB(cos(φAz)*cos(φBy))

It can be similarly shown that:

• y_a*y_o = RA*RB(cos(φAy)*cos(φBy)
• x_a*z_o= RA*RB(cos(φAx)*cos(φBz)
• z_a*x_o= RA*RB(cos(φAz)*cos(φBx)
• z_a*z_o= RA*RB(cos(φAz)*cos(φBz)
• x_a*y_o= RA*RB(cos(φAx)*cos(φBy)
• y_a*z_o= 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 i^v or with j^{^} to get j^v as these are π^{^} rotations. It can be achieved by 2 multiplications such as i^{^}*j^{^} = 1^{^}, i^{^}*1^{^} = j^v.

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