Editing Cross product (section)

=== Lagrange's identity ===
The relation
:<math>
  \left\| \mathbf{a} \times \mathbf{b} \right\|^2 \equiv
  \det \begin{bmatrix}
    \mathbf{a} \cdot \mathbf{a} & \mathbf{a} \cdot \mathbf{b} \\
    \mathbf{a} \cdot \mathbf{b}  & \mathbf{b} \cdot \mathbf{b}\\
  \end{bmatrix} \equiv
  \left\| \mathbf{a} \right\| ^2  \left\| \mathbf{b} \right\| ^2 - (\mathbf{a} \cdot \mathbf{b})^2
</math>

can be compared with another relation involving the right-hand side, namely [[Lagrange's identity]] expressed as<ref name=Boichenko>{{cite book |title=Dimension theory for ordinary differential equations |author1=Vladimir A. Boichenko |author2=Gennadiĭ Alekseevich Leonov |author3=Volker Reitmann |url=https://books.google.com/books?id=9bN1-b_dSYsC&pg=PA26 |page=26 |isbn=3-519-00437-2 |year=2005 |publisher=Vieweg+Teubner Verlag}}</ref>

:<math>
  \sum_{1 \le i < j \le n} \left( a_ib_j - a_jb_i \right)^2 \equiv
  \left\| \mathbf a \right\|^2 \left\| \mathbf b \right\|^2 - ( \mathbf{a \cdot b } )^2,
</math>

where '''a''' and '''b''' may be ''n''-dimensional vectors. This also shows that the [[Riemannian volume form]] for surfaces is exactly the [[Volume form|surface element]] from vector calculus. In the case where {{nowrap|1=''n'' = 3}}, combining these two equations results in the expression for the magnitude of the cross product in terms of its components:<ref name=Lounesto1>{{cite book |url=https://books.google.com/books?id=kOsybQWDK4oC&q=%22which+in+coordinate+form+means+Lagrange%27s+identity%22&pg=PA94 |author=Pertti Lounesto |page=94 |title=Clifford algebras and spinors |isbn=0-521-00551-5 |edition=2nd |publisher=Cambridge University Press |year=2001}}</ref>
 
:<math>\begin{align}
  \|\mathbf{a} \times \mathbf{b}\|^2
   &\equiv \sum_{1 \le i < j \le 3} (a_ib_j - a_jb_i)^2 \\
   &\equiv (a_1 b_2 - b_1 a_2)^2 + (a_2 b_3 - a_3 b_2)^2 + (a_3 b_1 - a_1 b_3)^2.
\end{align}</math>

The same result is found directly using the components of the cross product found from
:<math>\mathbf{a} \times \mathbf{b} \equiv \det \begin{bmatrix}
  \hat\mathbf{i} & \hat\mathbf{j} & \hat\mathbf{k} \\
  a_1 & a_2 & a_3 \\
  b_1 & b_2 & b_3 \\
\end{bmatrix}.</math>

In '''R'''<sup>3</sup>, Lagrange's equation is a special case of the multiplicativity {{nowrap|1={{abs|'''vw'''}} = {{abs|'''v'''}}{{abs|'''w'''}}}} of the norm in the [[Quaternion#Algebraic properties|quaternion algebra]].

It is a special case of another formula, also sometimes called Lagrange's identity, which is the three dimensional case of the [[Binet–Cauchy identity]]:<ref name=Liu/><ref name=Weisstein>by {{cite book |author=Eric W. Weisstein |chapter=Binet-Cauchy identity |title=CRC concise encyclopedia of mathematics |chapter-url=https://books.google.com/books?id=8LmCzWQYh_UC&pg=PA228 |page=228 |isbn=1-58488-347-2 |edition=2nd |year=2003 |publisher=CRC Press}}</ref>

:<math>
  (\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{c} \times \mathbf{d}) \equiv
  (\mathbf{a} \cdot \mathbf{c})(\mathbf{b} \cdot \mathbf{d}) - (\mathbf{a} \cdot \mathbf{d})(\mathbf{b} \cdot \mathbf{c}).
</math>

If {{nowrap|1='''a''' = '''c'''}} and {{nowrap|1='''b''' = '''d'''}}, this simplifies to the formula above.