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According to Wikipedia:

A rotation through an angle of $\theta$ around the axis defined by a unit vector $\vec{u} = (u_x, u_y, u_z) = u_x\mathbf{i} + u_y\mathbf{j} + u_z\mathbf{k}$ can be represented by a quaternion. This can be done using an extension of Euler's formula:

$$ \mathbf{q} = e^{\frac{\theta}{2}{(u_x\mathbf{i} + u_y\mathbf{j} + u_z\mathbf{k})}} = \cos \frac{\theta}{2} + (u_x\mathbf{i} + u_y\mathbf{j} + u_z\mathbf{k}) \sin \frac{\theta}{2}$$

Yet this identity is never proven. How does one do so?

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The same way de Moivre's is proven for $i$. The square roots of $-1$ in the quaternions $\mathbb{H}$ are precisely the unit imaginary quaternions $u$. (Exercise: prove this.) Therefore,

$$\begin{array}{ll} \exp(\theta u) & \displaystyle =1+\theta u+\frac{\theta^2}{2!}u^2+\frac{\theta^3}{3!}u^3+\cdots \\ & \displaystyle =\left(1-\frac{\theta^2}{2!}+\frac{\theta^4}{4!}-\cdots\right)+\left(\theta-\frac{\theta^3}{3!}+\frac{\theta^5}{5!}-\cdots\right)u \\ & =\cos(\theta)+\sin(\theta)u. \end{array} $$

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