R3 – Division – Definition
What is R3 Division? In the context of R3, division is defined as the inverse operation of multiplication. The result of dividing an operand by an operator is an expression such that when the operator is multiplied by this result, it yields the original operand. R3 division is a scalar operation that incorporates rotation through […]
R3 – Rotication
This post defines the new concept of the R3 rotication operation. It also provides examples. What is Rotication in R3? The rotication operation is grounded in the principles outlined in its axioms. Rotication consists of two steps. The first step is identical to rotation, resulting in a circular movement of a point. The second step takes […]
R3 – Multiplication – Cross Product
Cross product multiplication in R3 uses the same approach as in classical mathematics. The cross product of the points and is written as and is given by the formula: Unlike for dot-product, the Opposite Signs are used in multiplication. For example: Conclusion Try this example of R3 cross product multiplication with our online calculator. […]
R3 – Multiplication – Dot Product
Dot product multiplication in R3 uses the same approach as in classical mathematics. The dot product of the points and is written as and is given by the formula where a flip () precedes the Counters when the Opposite Value is or : For example: Euclidean Geometry Wikipedia describes dot product as follows: […]
R3 – Multiplication – Advanced
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R3 – Multiplication Use of Trigonometry
This post covers the use of trigonometry in R3 multiplication. It also describes a trigonometric formula for multiplication. Finding the angles φx, φy and φz R3 multiplication uses trigonometry when it uses the cosines of the angles φx, φy and φz in the multiplication formula later in this post. The following is a description of […]
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 […]