quaternion product | Physics Library

Quaternion Product in Scalar Vector Form

The component formula for multiplying two quaternions is correct but can look like a collection of unrelated signs. A more revealing form separates each quaternion into a scalar part and a three dimensional vector part. In that notation the Hamilton product becomes a compact combination of the ordinary dot product and cross product.

Let

p = p + p, q = q + q, w w

where p_w and q_w are real scalars and p and q are pure quaternions identified with ordinary three dimensional vectors. The Hamilton product is

pq = (pwqw − p ⋅ q) + (pwq + qwp + p × q).

(1)

Equivalently, writing a quaternion as the ordered pair (q_w,q),

(pw,p )(qw,q) = (pwqw − p ⋅ q, pwq + qwp + p × q).

(2)

Equation (2) is one of the most useful quaternion formulas in physics and engineering. It exposes the geometric content of Hamilton multiplication and makes the sign convention immediately visible: the PhysicsLibrary Hamilton product contains the term +p × q.

The later PhysicsLibrary attitude articles use passive coordinate transformations. That application convention does not change equation (2). Hamilton algebra and active or passive attitude interpretation are separate choices.

1 Convention declaration

PhysicsLibrary uses Hamilton multiplication,

ij = k, jk = i, ki = j,

with reversed products changing sign. Quaternions are displayed scalar first:

⌊ ⌋ qw q = q + q ←→ ||qx|| , w ⌈qy⌉ qz

(3)

where

q = qxi + qyj + qzk.

The product convention is determined by the algebra, not by whether software stores the scalar component first or last.

For comparison, a flipped multiplication used in some engineering literature reverses the Hamilton factors. Its scalar vector formula contains −p × q instead of the plus sign in equation (2). PhysicsLibrary does not use that multiplication unless an article explicitly states otherwise.

2 Product of two pure quaternions

The key result behind equation (2) is the product of two pure quaternions.

Let

a = axi + ayj + azk

and

b = bxi + byj + bzk.

Expand without changing the factor order:

ab = axbxi2 + axbyij + axbzik 2 + aybxji + aybyj + aybzjk + azbxki + azbykj + azbzk2.

Using the Hamilton basis products gives the scalar part

− (a b + a b + a b ) = − a ⋅ b, x x y y z z

and the vector part

(aybz − azby)i + (azbx − axbz)j + (axby − aybx)k,

which is exactly a × b.

Therefore

ab = − a ⋅ b + a × b.

(4)

Equation (4) is the central bridge between Hamilton quaternion multiplication and ordinary vector analysis.

3 Deriving the general scalar vector product

Write

p = p + p, q = q + q. w w

Distributivity gives

pq = pwqw + pwq + pqw + pq.

Real scalars commute with quaternion factors, so

pqw = qwp.

Using equation (4),

pq = − p ⋅ q + p × q.

Therefore

pq = p q − p ⋅ q w w + pwq + qwp + p × q.

Separating scalar and vector parts gives

Sc(pq) = pwqw − p ⋅ q,

(5)

and

Vec(pq) = pwq + qwp + p × q.

(6)

Together, equations (5) and (6) reproduce equation (2).

4 Connection to the component formula

Let

⌊ ⌋ ⌊ ⌋ px qx p = ⌈ py⌉ , q = ⌈qy⌉ . pz qz

The dot product is

p ⋅ q = pxqx + pyqy + pzqz,

and the cross product is

⌊ ⌋ pyqz − pzqy p × q = ⌈ pzqx − pxqz⌉ . pxqy − pyqx

Substitution into equations (5) and (6) gives

⌊ ⌋ pwqw − pxqx − pyqy − pzqz | pwqx + pxqw + pyqz − pzqy| [pq]PL = |⌈ p q − p q + p q + p q |⌉ . w y x z y w z x pwqz + pxqy − pyqx + pzqw

(7)

Thus the compact scalar vector formula and the four component formula are exactly equivalent.

5 Geometric meaning of the pure quaternion product

Equation (4),

ab = − a ⋅ b + a × b,

contains both the metric and oriented geometry of three dimensional Euclidean space.

If γ is the angle from a to b, then

a ⋅ b = ∥a ∥∥b∥ cosγ

and

∥a × b ∥ = ∥a∥∥b ∥sinγ.

Therefore the scalar part of ab contains the negative cosine information, while the vector part contains the oriented sine information.

This is why quaternion multiplication is naturally sensitive to order: the dot product is symmetric,

a ⋅ b = b ⋅ a,

but the cross product is antisymmetric,

b × a = − a × b.

6 Reversing the product

From equation (2),

pq = (pwqw − p ⋅ q, pwq + qwp + p × q ).

Reversing the order gives

qp = (pwqw − p ⋅ q, pwq + qwp − p × q ).

(8)

The scalar parts are identical. The difference lies entirely in the sign of the cross product.

Subtracting gives the commutator

[p,q] = pq − qp = 2p × q.

(9)

The commutator is therefore always a pure quaternion.

Adding the two products gives

pq + qp = 2 (pwqw − p ⋅ q ) + 2 (pwq + qwp ).

(10)

The cross product cancels from the symmetric combination.

7 When do two quaternions commute?

Equation (9) gives an immediate condition:

pq = qp ⇐ ⇒ p × q = 0.

Thus two quaternions commute exactly when their vector parts are parallel, antiparallel, or one vector part is zero.

For example, every pair of quaternions of the form

a + bu, c + du,

with the same fixed unit pure quaternion u, commute with each other. They form a two dimensional subalgebra isomorphic to the complex numbers.

8 Important special cases

Real scalar times a quaternion

If p = (a,0), then

pq = (aqw, aq ).

(11)

This is ordinary real scalar multiplication.

Two pure quaternions

If p = (0,p) and q = (0,q), then

pq = (− p ⋅ q, p × q) .

(12)

A pure quaternion squared

Set q = p. Because

p × p = 0,

equation (12) gives

2 2 p = − ∥p∥ .

(13)

Therefore every unit pure quaternion satisfies

2 u = − 1.

This result is fundamental to the later axis angle exponential.

Parallel pure quaternions

If b = λa, then

a × b = 0.

Hence

ab = − a ⋅ b,

(14)

which is purely real.

Perpendicular pure quaternions

a ⋅ b = 0,

then

ab = a × b,

(15)

which is pure.

For example,

ij = i × j = k.

Reversing the factors gives

ji = − k.

9 Hamilton and flipped multiplication

The sign of the cross product is a direct convention diagnostic.

PhysicsLibrary Hamilton multiplication uses

(pw,p )(qw,q) = (pwqw − p ⋅ q, pwq + qwp + p × q).

(16)

If another source defines a flipped product by reversing Hamilton factors,

p ⊗ q ≡ qp,

then its scalar vector form is

(pw,p) ⊗ (qw,q ) = (pwqw − p ⋅ q, pwq + qwp − p × q ).

(17)

Therefore

+p × q

identifies Hamilton multiplication and

− p × q

identifies the reversed product when all other definitions are held fixed.

This algebraic distinction must not be confused with the later choice between active and passive attitude interpretation. PhysicsLibrary retains equation (16) while using passive coordinate transformations for attitude.

10 Worked example 1: product from scalar and vector parts

Let

p = 2 + i − 2j + k, q = − 1 + 3i + 2k.

Then

⌊ ⌋ 1 pw = 2, p = ⌈ − 2⌉ , 1

and

⌊ ⌋ 3 q = − 1, q = ⌈ 0⌉ . w 2

The dot product is

p ⋅ q = 1 (3 ) + (− 2)(0) + 1 (2 ) = 5.

The cross product is

⌊ ⌋ − 4 p × q = ⌈ 1 ⌉ . 6

The scalar part is

pwqw − p ⋅ q = 2(− 1) − 5 = − 7.

The vector part is

⌊ ⌋ ⌊ ⌋ ⌊ ⌋ 3 1 − 4 p q + q p + p × q = 2⌈ 0⌉ − ⌈ − 2⌉ + ⌈ 1 ⌉ w w ⌊ 2⌋ 1 6 1 = ⌈3 ⌉. 9

Therefore

pq = − 7 + i + 3j + 9k.

(18)

This agrees with the component calculation in the preceding PhysicsLibrary article, quaternion definition and basic algebra.

11 Worked example 2: pure quaternion product

Let

⌊ ⌋ ⌊ ⌋ 1 0 a = ⌈2 ⌉, b = ⌈ 1⌉ . 0 3

Their dot product is

a ⋅ b = 2,

and their cross product is

⌊ 6 ⌋ ⌈ ⌉ a × b = − 3 . 1

Therefore

ab = − 2 + 6i − 3j + k.

(19)

Reversing the factors keeps the scalar part and reverses the vector part:

ba = − 2 − 6i + 3j − k.

(20)

12 Worked example 3: recovering the basis rule

Take

a = i, b = j.

Because the two unit vectors are perpendicular,

i ⋅ j = 0,

and

i × j = k.

Equation (4) gives

ij = k.

(21)

Thus the familiar Hamilton basis rule is contained directly in the scalar vector product.

13 Common pitfalls

Using the wrong sign on the cross product. PhysicsLibrary uses Hamilton multiplication, so the vector part contains +p × q.
Forgetting the minus sign on the dot product. For pure quaternions, ab = −a ⋅ b + a × b.
Treating quaternion multiplication as componentwise multiplication. The dot and cross products couple all three vector components.
Assuming the product of two pure quaternions is pure. Its scalar part is generally −a ⋅ b.
Assuming the product of two pure quaternions is real. Its vector part is generally a × b.
Interchanging factors. Reversing the factors reverses the cross product and therefore generally changes the quaternion.
Confusing storage order with multiplication convention. A scalar last array can still use Hamilton multiplication.
Changing the cross product sign because the attitude convention is passive. The passive PhysicsLibrary attitude convention does not alter Hamilton multiplication.

14 Exercises

The exercises are stated first so the article can be used for self study. Complete solutions follow afterward.

Derive the pure product. Starting from the Hamilton basis products, derive $ab = − a ⋅ b + a × b.$
Scalar vector product. Let $p = (2,[1,− 2,1]T ), q = (− 1,[3,0,2]T).$
Compute pq using equation (2).
Reverse the order. For the quaternions in Exercise 2, compute qp without repeating the complete calculation. Use the symmetry of the dot product and antisymmetry of the cross product.
Pure quaternion square. Prove from equation (4) that $a2 = − ∥a∥2.$
Unit pure quaternion. If ∥u∥ = 1, prove that u² = −1.
Perpendicular vectors. Let a ⋅ b = 0. Show that $ab = a × b$
and
$ba = − a × b.$
Parallel vectors. If b = λa, show that ab is purely real.
Commutator. Derive $[p,q] = 2p × q.$
Commuting quaternions. Use Exercise 8 to prove that p and q commute if and only if their vector parts are parallel, antiparallel, or one is zero.
Convention diagnostic. A source states $(pw,p ) ⋆ (qw,q) = (pwqw − p ⋅ q, pwq + qwp − p × q) .$
Does ⋆ agree with PhysicsLibrary Hamilton multiplication? Explain.

15 Solutions

1. Derive the pure product

Expand

a = axi + ayj + azk

and

b = bxi + byj + bzk.

The square terms contribute

− axbx − ayby − azbz = − a ⋅ b.

The mixed basis terms contribute

(aybz − azby)i + (azbx − axbz)j + (axby − aybx)k,

which is a × b. Therefore

ab = − a ⋅ b + a × b.

2. Scalar vector product

For

⌊ ⌋ 1 pw = 2, p = ⌈ − 2⌉ , 1

and

⌊ 3⌋ ⌈ ⌉ qw = − 1, q = 0 , 2

we have

p ⋅ q = 5

and

⌊ ⌋ − 4 p × q = ⌈ 1 ⌉ . 6

Thus the scalar part is −7, and the vector part is

⌊ ⌋ 1 2q − p + p × q = ⌈ 3⌉ . 9

Therefore

pq = − 7 + i + 3j + 9k.

3. Reverse the order

The scalar part remains −7. Reversing the factors changes

p × q

− p × q.

Therefore

qp = − 7 + 9i + j − 3k.

4. Pure quaternion square

Set b = a in equation (4):

2 a = − a ⋅ a + a × a.

Since

a ⋅ a = ∥a∥2

and

a × a = 0,

we obtain

2 2 a = − ∥a∥ .

5. Unit pure quaternion

If ∥u∥ = 1, Exercise 4 gives immediately

2 u = − 1.

6. Perpendicular vectors

a ⋅ b = 0,

then

ab = a × b.

Reversing the factors gives

ba = b × a = − a × b.

7. Parallel vectors

b = λa,

then

a × b = λa × a = 0.

Therefore

ab = − a ⋅ b,

which is purely real.

8. Commutator

Use equations (2) and (8). The scalar parts cancel, as do the terms p_wq + q_wp. The cross terms give

p × q − (− p × q) = 2p × q.

Hence

[p,q] = 2p × q.

9. Commuting quaternions

The quaternions commute exactly when

[p,q] = 0.

By Exercise 8 this is equivalent to

p × q = 0.

A cross product vanishes exactly when the two vectors are parallel, antiparallel, or one is zero.

10. Convention diagnostic

The operation ⋆ does not agree with PhysicsLibrary Hamilton multiplication because its vector part contains

− p × q.

PhysicsLibrary Hamilton multiplication contains

+p × q.

The sign difference identifies a reversed or flipped multiplication when the remaining definitions are held fixed. It is not caused by scalar first or scalar last storage, and it is not caused by the later passive attitude interpretation.

16 What comes next

The scalar vector form makes several later quaternion identities almost automatic.

For the conjugate,

q∗ = qw − q,

equation (2) immediately shows why

qq∗ = q∗q

is purely real: the cross product of q with itself vanishes. That result leads directly to the quaternion norm and inverse.

The later orientation articles use the same Hamilton product to construct passive frame transformations. No algebraic sign in equation (2) is changed when the application changes from general quaternion algebra to passive attitude representation.

17 Sources and historical notes

Hamilton’s quaternion algebra contains both scalar and vector parts of products. Later quaternion and vector analysis texts made the relationship to dot and cross products increasingly explicit. Joly’s A Manual of Quaternions and Macfarlane’s Vector Analysis and Quaternions are useful public domain historical references for this decomposition.

The modern PhysicsLibrary notation uses ordinary dot and cross product symbols and scalar first quaternion components. The exposition and worked examples in this entry are newly organized for physics and engineering use.

References

[1] W. R. Hamilton, Elements of Quaternions, 2nd ed., edited by C. J. Joly, Longmans, Green, and Co., 1899. Public domain historical source. Internet Archive scan

[2] C. J. Joly, A Manual of Quaternions, Macmillan and Co., London, 1905. Public domain historical source. Internet Archive scan

[3] A. Macfarlane, Vector Analysis and Quaternions, John Wiley and Sons, New York, 1906. Public domain historical source. Project Gutenberg edition

[4] H. Sommer, I. Gilitschenski, M. Bloesch, S. Weiss, R. Siegwart, and J. Nieto, “Why and How to Avoid the Flipped Quaternion Multiplication,” Aerospace, vol. 5, no. 3, article 72, 2018. Published under CC BY 4.0. Publisher article

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