Maths - Dual Vectors


If you are not familiar with this subject you may like to look at the following pages first:


Dual numbers have the form:

a + w b


w2 = 0

In the case where w2 = 0 we indicate this by replacing w with ε to give:

a + ε b

for dual vectors 'a' and 'b' can be vectors.

This is similar to the way that quaternions can be represented as a real part (scalar) and an imaginary part which is a vector.

For dual vectors either or both parts may be vectors.

Relationship to 3D Geometric/Clifford algebra

What is the relationship between dual vectors and 3D Geometric/Clifford algebra?

dual vectors has:

3D Geometric/Clifford algebra has:

So could 3D Geometric algebra be a superset of dual vectors?

In 3D Geometric algebra squaring either of the vectors will give a scalar value, so if we dont have a scalar value then perhaps squaring them will give zero?

Applications of dual vectors

Dual vectors can be used to represent 'screw displacement' that is, a representation of velocity of a solid body, a combination of linear and angular velocity. (see kinimatics page on this subject).


The components of a and d are known as plucker coordinates.

To represent this using the dual vector a + w b

then b = a x d


Further Reading

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see also:


Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

cover us uk de jp fr ca Quaternions and Rotation Sequences.


Terminology and Notation

Specific to this page here:


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