Quaternion

From Projective Geometric Algebra
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A quaternion is an operator that performs a rotation about the origin in 3D space. Conventionally, a quaternion $$\mathbf q$$ is written as

$$\mathbf q = q_w + q_x \mathbf i + q_y \mathbf j + q_z \mathbf k$$ ,

where the "imaginary" units $$\mathbf i$$, $$\mathbf j$$, and $$\mathbf k$$ all square to $$-1$$.

The set of quaternions corresponds to a part of the three-dimensional geometric algebra. Specifically, the quaternions are exactly the elements that can be expressed as the sum of a scalar and a 3D bivector. Making the identifications

$$\mathbf i = -\mathbf e_23$$
$$\mathbf j = -\mathbf e_31$$
$$\mathbf k = -\mathbf e_12$$ ,

a quaternion $$\mathbf q$$ can be written as

$$\mathbf q = q_w - q_x \mathbf e_23 - q_y \mathbf e_31 - q_z \mathbf e_12$$ .

See Also