Quaternion Multiplication: Expanding and Simplifying

[flash]

Homework Statement

Expand and simplify the product of two quaternions:

(3 + 2i + 3j + 4k)(3 + 3i + 2j + 5k)

Justify your response.

The Attempt at a Solution

I have done this by expanding brackets normally, keeping the ijk's in the same order because the multiplication is not commutative. I have the correct answer, I just can't think how to justify it. Why should expanding brackets necessarily work?

 

[cristo]

Why should expanding brackets necessarily work?

Isn't that how multiplication of the quaternions is defined?

 

[D H]

Why should expanding brackets necessarily work?

It works necessarily because quaternion multiplication distributes over quaternion addition per the definitions of quaternion multiplication and addition.

 

[flash]

Thanks guys, that's all I needed.