R3 – Multiplication – Simple
This post looks at some simple R3 multiplication of Opposite Values together and using counters. The examples show the distributivity of R3 multiplication. However, they also show that R3 multiplication is not associative or commutative. R3 Unitary Multiplication Table * 1^ 1v i^ iv j^ jv 1^ 1^ 1v j^ jv iv i^ 1v 1^ […]
R3 – Multiplication – Definition
What is R3 Multiplication? Multiplication in R3 is a scalar operation that incorporates rotation through Opposite Types and Signs. It follows a “times-add” process, with the Opposite Sign of the result determined by the multiplication table. It differs from R1 and R2 multiplication because it now brings in the concept of rotation in real space compared […]
R3 – Complex Rotations – Example 1
Up to this point, we’ve focused on unitary rotations around a single axis on the unit sphere. Now, let’s delve into more complex rotations in R3, where the operand can be any combination of Opposite Values with any degree of rotation. This post introduces the Wave Number Rotation formula, based on Rodrigues’ formula for calculating […]
R3 – Rotation – Hyperspherical Coordinates
This post covers the formula for hyperspherical coordinates in R3. Examples of rotation using the formula are given. Hyperspherical Coordinates Wikipedia defines the hyperspherical coordinate system as: ‘a coordinate system in an n-dimensional Euclidean space which is analogous to the spherical coordinate system defined for 3-dimensional Euclidean space, in which the coordinates consist of a radial coordinate r, and n […]
R3 – Rotation – Spherical Coordinates
This post covers the formula for spherical coordinates in R3. Examples of rotation using the formula are given. Cartesian Coordinates The Cartesian coordinates of a point are based on the intersection of perpendicular lines from the point with the x, y and z-axes. For example: the Cartesian coordinates at a point on the surface of […]
R3 – Rotation – Multiplication Table
Orthogonal Rule The layout of the R3 axes is defined as part of the Wave Number axioms. The Axioms of Rotation provide the Orthogonal rule which states that multiplication by a unitary is the equivalent of rotation around the unitary’s axis by 90o. This allows for the derivation of the R3 unitary multiplication table, shown below, from rotations […]
R3 – Rotation – Examples
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R3 – Rotation – Simple Examples
Tait-Bryan Angles This post looks at simple R3 rotations that consist of 90o rotations around a single axis on the unit sphere. A 90^o axis-angle rotation around the x-axis is a z-y’-x’’ rotation using the Tait-Bryan angles of φ = 90^o, θ = 0o and ψ= 0o. Rotation of 90^o around the x-axis moves the […]
R3 – Addition
In R3, addition in the Wave Number system is straightforward. Opposite Values with Opposite Signs cancel each other out through interference. See the examples below: Examples of R3 Addition Conclusion Try these examples of R3 addition with our online calculator. Next: Rotation Previous: R2 Logs
Rotation Sphere
R3 – Rotation Definition and Quaternions What is Rotation in R3? The R3 rotation operation is a fundamental aspect of the Wave Number system. The facts expressed in its axioms support this. A rotation moves a point, represented by Opposite Values, to a new location, also represented by Opposite Values, through circular movement around an […]