Give product of two quaternions.

Question

Compute the product of the quaternions $(a+ bi +aj)(ai + ck)$ with $a,b,c\in \mathbb Z.$

But, answer shown on Wolfram Alpha is different?

Answer 1

It sounds like you just entered into Wolfram Alpha without telling it to work in quaternions. It usually understands you mean the imaginary unit when you type i, but for j and k it usually treats them as variables not unit quaternions.

If you tell it you are in quaternions by putting an explicit call to that function, then you can get the objects you intended.

https://www.wolframalpha.com/input?i=Quaternion%5B1%2C-2%2C1%2C0%5D*Quaternion%5B0%2C1%2C0%2C-4%5D

Once you have the factors as quaternions through entering them within Quaternion[...] then you can multiply them.