Quaternions and the geometry of R3

Because the agent allotment of a quaternion is a agent in R3, the geometry of R3 is reflected in the algebraic anatomy of the quaternions. Many operations on vectors can be authentic in agreement of quaternions, and this makes it accessible to administer quaternion techniques wherever spatial vectors arise. For instance, this is accurate in electrodynamics and 3D computer graphics.

For the butt of this section, i, j, and k will denote both imaginary18 base vectors of H and a base for R3. Notice that replacing i by −i, j by −j, and k by −k sends a agent to its accretion inverse, so the accretion changed of a agent is the aforementioned as its conjugate as a quaternion. For this reason, alliance is sometimes alleged the spatial inverse.

Choose two abstruse quaternions p = b1i + c1j + d1k and q = b2i + c2j + d2k. Their dot artefact is

This is according to the scalar locations of p*q, qp*, pq*, and q*p. (Note that the agent locations of these four articles are different.) It aswell has the formulas

The cantankerous artefact of p and q about to the acclimatization bent by the ordered base i, j, and k is

(Recall that the acclimatization is all-important to actuate the sign.) This is according to the agent allotment of the artefact pq (as quaternions), as able-bodied as the agent allotment of −q*p*. It aswell has the formula

In general, let p and q be quaternions (possibly non-imaginary), and write

where ps and qs are the scalar locations of p and q and and are the agent locations of p and q. Then we accept the formula

This shows that the noncommutativity of quaternion multiplication comes from the multiplication of authentic abstruse quaternions. It aswell shows that two quaternions drive if and alone if their agent locations are collinear.

For added accession on clay three-dimensional vectors application quaternions, see quaternions and spatial rotation.

editMatrix representations

Just as circuitous numbers can be represented as matrices, so can quaternions. There are at atomic two means of apery quaternions as matrices in such a way that quaternion accession and multiplication accord to cast accession and cast multiplication. One is to use 2×2 circuitous matrices, and the added is to use 4×4 absolute matrices. In anniversary case, the representation accustomed is one of a ancestors of linearly accompanying representations. In the analogue of abstruse algebra, these are injective homomorphisms from H to the cast rings M2(C) and M4(R), respectively.

Using 2×2 circuitous matrices, the quaternion a + bi + cj + dk can be represented as

This representation has the afterward properties:

Complex numbers (c = d = 0) accord to askew matrices.

The barometer of a quaternion (the aboveboard basis of a artefact with its conjugate, as with circuitous numbers) is the aboveboard basis of the account of the agnate matrix.19

The conjugate of a quaternion corresponds to the conjugate alter of the matrix.

Restricted to assemblage quaternions, this representation provides an isomorphism amid S3 and SU(2). The closing accumulation is important for anecdotic circuit in breakthrough mechanics; see Pauli matrices.

Using 4×4 absolute matrices, that aforementioned quaternion can be accounting as

In this representation, the conjugate of a quaternion corresponds to the alter of the matrix. The fourth ability of the barometer of a quaternion is the account of the agnate matrix. Circuitous numbers are block askew matrices with two 2×2 blocks.