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 […]
R2 – Logs
This post covers the definition of R2 logs, logarithm formulae and provides some examples. Definition Wikipedia defines logs as follows: ‘In mathematics, logarithm (log) is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x.’ Logs are useful in expressing […]
R2 – Exponentiation – Fractions, Flip Sign and Rules
This post covers the rules of R2 exponentiation. It also describes the use of fractions and the flip sign with exponentiation along with some examples. Fractions Exponents in fractions represent multiplication roots. As described in an earlier post, the root of an expression can be expressed as the root of the counter multiplied by the […]
R2 – Exponentiation – Definition
Introduction This post covers the definition of R2 Exponentiation in Wave Numbers. It also includes reciprocals and examples. In Wikipedia, exponentiation is defined as follows: ‘Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent or power n, and pronounced as “b raised to the power of n”. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that […]
R2 – Roots – Advanced
This post covers the use or De Moivre’s formula as an advanced method of calculating roots in R2. Some examples are provided. Formula The roots of any Opposite Value can be calculated using the formula below which is based on De Moivre’s formula. This formula relies on the fact that the root of an Opposite […]