# Definition:Quaternion/Multiplication

The product of two quaternions $\mathbf x_1 = a_1 \mathbf 1 + b_1 \mathbf i + c_1 \mathbf j + d_1 \mathbf k$ and $\mathbf x_2 = a_2 \mathbf 1 + b_2 \mathbf i + c_2 \mathbf j + d_2 \mathbf k$ is defined as:
 $\displaystyle \mathbf x_1 \mathbf x_2$ $:=$ $\displaystyle \left({a_1 a_2 - b_1 b_2 - c_1 c_2 - d_1 d_2}\right) \mathbf 1$ $\displaystyle$  $\, \displaystyle + \,$ $\displaystyle \left({a_1 b_2 + b_1 a_2 + c_1 d_2 - d_1 c_2}\right) \mathbf i$ $\displaystyle$  $\, \displaystyle + \,$ $\displaystyle \left({a_1 c_2 - b_1 d_2 + c_1 a_2 + d_1 b_2}\right) \mathbf j$ $\displaystyle$  $\, \displaystyle + \,$ $\displaystyle \left({a_1 d_2 + b_1 c_2 - c_1 b_2 + d_1 a_2}\right) \mathbf k$