R3 – Division – Examples
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R3 – Division – Simultaneous Equations
Why Simultaneous Equations are Needed According to the theorem of multiplicative inverses, R3 lacks multiplicative inverses. Therefore, the Wave Number system employs an alternative method to compute the result of division in R3, using simultaneous equations. Consider the following equation: To obtain the result of the division, you need to solve the following equation: The […]
R3 – Reciprocals and Multiplicative Inverses
This post covers multiplicative inverses and the reciprocals of Opposite Values in R3. Need for Multiplicative Inverses As stated in the theorem in an earlier post, multiplicative inverses are not available in R3. Multiplicative inverses do not exist, as multiplication is not commutative in R3. The main use of the multiplicative inverse is to avoid […]
R3 – Division – Self Division and Unitaries
This post covers self division in R3. It looks at the cases where the result of division is a unitary and where it is not. Self Division In quaternions, R1 and R2, the division of an expression by itself results in the answer 1^. For example in R2: However, in Wave Numbers R3, division of […]
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 […]