diff --git a/00.brainstorming-pedagogical-teams/40.collection-existing-pedagogical-content/05.classical-mechanics/vector-analysis/textbook.fr.md b/00.brainstorming-pedagogical-teams/40.collection-existing-pedagogical-content/05.classical-mechanics/vector-analysis/textbook.fr.md
index 19e2f623a..d31a331df 100644
--- a/00.brainstorming-pedagogical-teams/40.collection-existing-pedagogical-content/05.classical-mechanics/vector-analysis/textbook.fr.md
+++ b/00.brainstorming-pedagogical-teams/40.collection-existing-pedagogical-content/05.classical-mechanics/vector-analysis/textbook.fr.md
@@ -509,10 +509,30 @@ tenseur de courbure, tenseur énergie-impulsion, ...
 
 ##### Componentes de un producto vectorial en base ortonormal / Composantes d'un produit vectoriel dans une base orthonormée / Components of a vector product in an orthonormal basis
 
-"$`(\vec{e_1},\vec{e_2},...,\vec{e_n})`$ est une base orthonormée.
+$`(\vec{e_1},\vec{e_2},...,\vec{e_n})`$ est une base orthonormée
 $`\quad\Longrightarrow`$
 $`\displaystyle\quad\forall \overrightarrow{U}\in\mathcal{P}\quad \overrightarrow{U}=\sum_{i=1}^n\;U_i\cdot\vec{e_i}`$ 
 $`\displaystyle\quad\forall \overrightarrow{V}\in\mathcal{P}\quad \overrightarrow{V}=\sum_{i=1}^n\;V_i\cdot\vec{e_i}`$  
+
+For the expression of a vector $`\vec{U}`$ in the base $`(\vec{e_1},\vec{e_2},...,\vec{e_n})`$,
+we should use :
+
+$`\overrightarrow{U}=\left(\begin[array](l)U_1//U_2//U_3)\end[array]\right)`$
+
+
+
+
+méthode des produits en croix :
+
+
+http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=102-03-04
+
+$`\overrightarrow{U}=\begin\left
+
+
+method similar to the sum used to obtain the determinant of a matrix :
+
+
 **$`\displaystyle\quad\overrightarrow{U}\cdot\overrightarrow{V}=U_1\,V_1 + U_2\,V_2 + ... + U_n\,V_n = \sum_{i=1}^n\;U_i\,V_i`$**