Appendix P Longitudinal and transversal projection of vectors

A vector $𝒂$ can be split into two parts: the longitudinal part ${𝒂}_{\parallel }$, which is parallel to another vector $𝒃$ and the transversal part ${𝒂}_{⟂}$, which is perpendicular to $𝒃$. The length of the longitudinal part $a_{\parallel }$ and the transversal part $a_{⟂}$ can be computed from geometry (see figure LABEL:fig:projection)

	${\displaystyle \frac{a_{\parallel }}{a}}=\cos \alpha$	$\Rightarrow a_{\parallel }=a\cos \alpha ={\displaystyle \frac{ab\cos \alpha }{b}}={\displaystyle \frac{𝒂\cdot 𝒃}{b}},$		(P.1)
	${\displaystyle \frac{a_{⟂}}{a}}=\sin \alpha$	$\Rightarrow a_{⟂}=a\sin \alpha ={\displaystyle \frac{ab\sin \alpha }{b}}={\displaystyle \frac{\left|𝒂\times 𝒃\right|}{b}}.$		(P.2)

But from the Pythagorean theorem we can get another expression for the length of the transversal part

$a^2=a_{\parallel }^2+a_{⟂}^2$

$\Rightarrow a_{⟂}^2=a^2-a_{\parallel }^2=a^2-{\displaystyle \frac{{(𝒂\cdot 𝒃)}^2}{b^2}}.$

(P.3)

Substituting equation (P.2) in equation (P.3) we get

${\displaystyle \frac{{(\left|𝒂\times 𝒃\right|)}^2}{b^2}}=a^2-{\displaystyle \frac{{(𝒂\cdot 𝒃)}^2}{b^2}},$

which leads us to the following expression for the square of the norm of the cross product

${\left|𝒂\times 𝒃\right|}^2={(ab)}^2-{(𝒂\cdot 𝒃)}^2.$

(P.4)