Maths - Clifford Algebra - 2D Arithmetic - Inner and Outer product

In addition to the geometric product there are two more types of multiplication used in Geometric Algebra. We want to extend and generalise the 'dot' and 'cross' products used in 3D vector algebra to be applied in any number of dimensions, in this case, 2D.

Inner product by a vector reduces the grade of a multivector. It is related to the dot product.

Outer product by a vector increases the grade of a multivector. It is related to the cross product.

Outer product

We want the outer product to be an extension of the vector cross product:

x = Ay * Bz - By * Az
y = Az * Bx - Bz * Ax
z = Ax * By - Bx * Ay

This gives a bivector which is mutually perpendicular to the vectors being multiplied. But how do we extend this so that we can multiply non-vectors? There are different definitions for the outer product so we need to be careful, if you use other websites or books you may find that they use different definitions. Here is the full 2D Outer multiplication table:

a^b
b.e b.e1 b.e2 b.e12
a.e e e1 e2 e12
a.e1 e1 0 e12 0
a.e2 e2 -e12 0 0
a.e12 e12 0 0 0

This has the following nice properties:

It is relatively easy to derive using the following rules:

By starting with the vector cross product terms we can derive the remaining terms in the table as follows:

e1 ^ e1 = 0
e2 ^ e2 = 0

because we have chosen to let vectors square to zero as with vector cross product.

e1 ^ e2 = e12
e2 ^ e1 = -e12

As with vector cross product, vectors anticommute, we can also derive this because the vectors will cancel out when a general vector is squared: (a e1 + b e2)2 =0.

e1 ^ e12 =0
e2 ^ e12 = 0
e12 ^ e1 = 0
e12 ^ e2 = 0

these are derived from the results above using these rules.

e12 ^ e12 = 0

these are derived from the results above using these rules.

Inner product.

We want the outer product to be an extension of the vector dot product:

A • B = Ax * Bx + Ay * By

This gives a scalar which depends on the angle between the vectors being multiplied, it is zero if the vectors are perpendicular and zero if the vectors are parallel. But how do we extend this so that we can multiply non-vectors? There are different definitions for the inner product so we need to be careful, if you use other websites or books you may find that they use different definitions. Here we will use the 'semi-commutative inner product' which has the following 2D Inner multiplication table:

a•b
b.e b.e1 b.e2 b.e12
a.e 0 0 0 0
a.e1 0 e 0 e2
a.e2 0 0 e -e1
a.e12 0 -e2 e1 -e

This has the following nice properties:

By starting with the vector cross product terms we can derive the remaining terms in the table as follows:

e1 • e1 = e
e2 • e2 = e
e2 • e2 = e

because we have chosen to let vectors square to zero as with vector cross product.

e1 • e2 = 0
e2 • e1 = 0

 

As with vector cross product vectors anticommute, we can also derive this because the vectors will cancel out when a general vector is squared: (a e1 + b e2)2 =a2 + b2

e1 • e12 =e2

e2 • e12 = -e2 • e21 = -e1

The result of inner multiplying a vector base by a bivector base depends on weather they contain a common base,

if they do, make the bases adjacent then the common base cancels out: e1 • e12 =e2

if they don't the result is zero: e1•(e2^e3)=0

Same thing for inner multiplying by a tri-vector.

e12 • e1 = -e21 • e1 = -e2
e31 • e1 = e3
e23 • e1 = 0
e123 • e1 =e231 • e1 = e23

e12 • e2 = e1
e31 • e2 = 0
e23 • e2 = -e32 • e2 = -e3
e123 • e2 = -e132 • e2 = -e13 =e31

e12 • e3 = 0
e31 • e3 = -e13 • e3 = -e1
e23 • e3 = e2
e123 • e3 = e12

 

e12 • e12 = -e21 • e12 = -e

 

When multiplying two bivectors, if the bivectors have the same bases then we need to reverse one of them to get adjacent terms together, then we can cancel out terms, so we always get -e.

If the bivectors are different the result will always be zero.

When multiplying a bivector by a trivector we reverse as necessarily to cancel out two of the terms to leave a vector.

Identities

The following identities relate the inner, outer and geometric products of vectors (grade one multivector) :

a•b = ½ (ab + ba) This is symmetrical (a•b = b•a)
a^b = ½ (ab - ba) This is anti-symmetrical (a^b = - b^a)
a * b = a•b + a^b  

Where:

We can extend this to the multiplication of a vector by a general multivector as follows:

a•K = ½ (aK + (-1)k+1Ka)

a^K = ½ (aK + (-1)k Ka)

a*K = a•K + a^K

Where k is the grade of K. The (-1)k factor alternates the sign as follows:

grade k (-1)k (-1)k+1 a•K = ½ (aK + (-1)k+1Ka) a^K = ½ (aK + (-1)k Ka)
0 (scalar) 1 -1 a•K = ½ (aK - Ka) = 0 a^K = ½ (aK + Ka) = aK
1 (vector) -1 1 a•K = ½ (aK + Ka) = aK a^K = ½ (aK - Ka) = 0
2 (bivector) 1 -1 a•K = ½ (aK - Ka) = 0 a^K = ½ (aK + Ka) = aK

However we have not yet created a general expression for any grade of multivector, for instance a bivector times a bivector, which does not follow the above pattern.

A<2> ^ B<2> = 0

Can anyone help with a more general expression.

Other identities for •

(a^b)•(c^d) = a•(b•(c^d))= b•(c^d)•a from Hestenes (New Foundations of Classical Mechanics) pp47 exercise 2.1

The inner product is not associative: a•(b•c) may not equal (a•b)•c


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flag flag flag flag flag flag Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (Fundamental Theories of Physics). This book is intended for mathematicians and physicists rather than programmers, it is very theoretical. It covers the algebra and calculus of multivectors of any dimension and is not specific to 3D modelling.

 

 

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