July 22, 2024
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R3 – Quaternion Comparison

Comparison of Wave Number and Quaternion Multiplication Tables:

An earlier post makes a detailed comparison of the Wave Number Multiplication table and the Quaternion Multiplication table.

The only difference occurs when Quaternions are multiplied together, as in i*i, = -1, but in R3, 1^*1^ = 1^

In R3, multiplying 1^ and 1^ = 1^ and multiplying 1^ and 1v = 1v, leaving the operand unchanged.

Rotation Comparison

Another post explains the calculation of the rotation of 60o^ ↺(4v+ 5iv + 6j^)  (1v + 2i^ + 3jv) =  (0.419^ + 0.003i^ +  3.718jv) and compares it to the equivalent Quaternion rotation. The result of Quaternion rotation is not in 3d, so a final step is required to convert the Quaternion result into an R3 vector. This is done by multiplying the result with the conjugate of the Quaternion of Rotation to get the R3 vector. The Quaternion and Wave Number results match.

Wave Number and Quaternion Efficiency

A third post gives a comparison between the efficiency of rotation of Wave Numbers and Quaternions. This comparison makes two conclusions. The first is that the efficiency of Wave Numbers is greater. The second is that the Wave Number representation is clearer than Quaternions as Wave Numbers are always in 3d space.

Squares Comparison with Quaternions

A final comparison looks at squares in R3 Wave Numbers and Quaternions.

Quaternions are in the form (a + bi + cj + dk). A Quaternion with just a value for a is a scalar whereas a Quaternion where a = 0 is a vector or pure Quaternion. They represent vectors in real R3 space. For example: (2i – 3j + 4k) is a pure Quaternion and -29 is a scalar.

In Quaternions the square of a pure Quaternion is a Quaternion scalar. So, (2i – 3j + 4k)(2i – 3j + 4k) = -29. This is a direct result of the rules of Quaternion multiplication.

In Wave Numbers the equivalent squaring of (2^ + 3iv + 4j^)*(2^ + 3iv + 4j^) = (4^ + 9iv + 16j^).

Intuitively, the Wave Number result seems more realistic. Wave Numbers gives a result that is a point of greater magnitude in 3d space. Quaternions gives a scalar which is not in 3d space.

Conclusion

It is now evident how the Wave Numbers system works in R3. No such system is available in classical maths. Wave Numbers reside naturally in R3 without the 4d complexities that Quaternions bring.

Try out the online calculator to check out any of these calculations.

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