Editing Screw theory (section)

== Algebra of screws ==
Let a ''screw'' be an ordered pair 
: <math> \mathsf{S} = (\mathbf{S}, \mathbf{V}), </math>
where {{math|'''S'''}} and {{math|'''V'''}} are three-dimensional real vectors.  The sum and difference of these ordered pairs are computed componentwise.  Screws are often called ''dual vectors''.

Now, introduce the ordered pair of real numbers {{nowrap|1=â = (''a'', ''b'')}}, called a [[dual number|''dual scalar'']].  Let the addition and subtraction of these numbers be componentwise, and define multiplication as 
<math display="block"> \hat{\mathsf{a}}\hat{\mathsf{c}}=(a, b)(c, d) = (ac, ad + bc). </math>
The multiplication of a screw {{nowrap|1=S = ('''S''', '''V''')}} by the dual scalar {{nowrap|1=â = (''a'', ''b'')}} is computed componentwise to be,
<math display="block"> \hat{\mathsf{a}}\mathsf{S}  = (a, b)(\mathbf{S}, \mathbf{V}) = (a \mathbf{S}, a \mathbf{V} +b \mathbf{S}).</math>

Finally, introduce the dot and cross products of screws by the formulas: 
<math display="block"> \mathsf{S}\cdot \mathsf{T} = (\mathbf{S}, \mathbf{V})\cdot (\mathbf{T}, \mathbf{W})  = (\mathbf{S}\cdot\mathbf{T},\,\, \mathbf{S}\cdot\mathbf{W} +\mathbf{V}\cdot\mathbf{T}), </math>
which is a dual scalar, and  
<math display="block"> \mathsf{S}\times \mathsf{T} = (\mathbf{S}, \mathbf{V})\times (\mathbf{T}, \mathbf{W}) = (\mathbf{S}\times \mathbf{T},\,\, \mathbf{S}\times \mathbf{W} +\mathbf{V}\times \mathbf{T}),</math>
which is a screw. The dot and cross products of screws satisfy the identities of vector algebra, and allow computations that directly parallel computations in the algebra of vectors.

Let the dual scalar {{nowrap|1=ẑ = (''φ'', ''d'')}} define a ''dual angle'', then the infinite series definitions of sine and cosine yield the relations
<math display="block"> \sin \hat{\mathsf{z}} = (\sin\varphi , d \cos\varphi), \,\,\,  \cos\hat{\mathsf{z}} = (\cos\varphi ,- d \sin\varphi),</math>
which are also dual scalars. In general, the function of a dual variable is defined to be {{nowrap|1={{itco|''f''}}(ẑ) = ({{itco|''f''}}(''φ''), ''df''′(''φ''))}}, where {{itco|''df''}}′(''φ'') is the derivative of&nbsp;{{itco|''f''}}(''φ'').

These definitions allow the following results:
* Unit screws are [[Plücker coordinates]] of a line and satisfy the relation <math display="block"> |\mathsf{S}| = \sqrt{\mathsf{S} \cdot \mathsf{S}} = 1; </math>
* Let {{nowrap|1=ẑ = (''φ'', ''d'')}} be the dual angle, where ''φ'' is the angle between the axes of S and T around their common normal, and ''d'' is the distance between these axes along the common normal, then <math display="block"> \mathsf{S} \cdot \mathsf{T} = \left|\mathsf{S}\right| \left|\mathsf{T}\right| \cos\hat{\mathsf{z}}; </math>
* Let N be the unit screw that defines the common normal to the axes of S and T, and {{nowrap|1=ẑ = (''φ'', ''d'')}} is the dual angle between these axes, then <math display="block"> \mathsf{S} \times \mathsf{T} = \left|\mathsf{S}\right| \left|\mathsf{T}\right| \sin\hat{\mathsf{z}} \mathsf{N}. </math>