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 b17024811..a5b463cdc 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
@@ -518,20 +518,24 @@ we shouldn't we use (http://www.electropedia.org/iev/iev.nsf/display?openform&ie
 $`\overrightarrow{U}=\begin{pmatrix}U_1\\U_2\\U_3\end{pmatrix}`$ or $`\overrightarrow{U}=\begin{bmatrix}U_1\\U_2\\U_3\end{bmatrix}`$
 instead of $`\overrightarrow{U}=\left|\begin{array}{l}U_1\\U_2\\U_3\end{array}\right.`$ as we do at INSA ?
 
-c
+* [ES]  <br>
 [FR] méthode des produits en croix :<br>
+[EN] <br>
 $`\forall\overrightarrow{U}=\begin{pmatrix}U_1\\U_2\\U_3\end{pmatrix}`$
 $`\quad\forall\overrightarrow{V}=\begin{pmatrix}U_1\\U_2\\U_3\end{pmatrix}`$
 $`\quad\vec{U}\land\vec{V}=\begin{pmatrix}U_1\\U_2\\U_3\end{pmatrix}\land\begin{pmatrix}V_1\\V_2\\V_3\end{pmatrix}`$
-$`\quad=±,\begin{pmatrix}U_2 V_3 - U_3 V_2\\U_3 V_1 - U_1 V_3\\U_1 V_2 - U_2 V_1\end{pmatrix}`$
+$`\begin{pmatrix}U_2 V_3 - U_3 V_2\\U_3 V_1 - U_1 V_3\\U_1 V_2 - U_2 V_1\end{pmatrix}`$
+$`=U_1V_1\,\overrightarrow{e_3}+U_2V_3\,\overrightarrow{e_1}+U_3V_1\,\overrightarrow{e_2}`$
+$`- U_1V_3\,\overrightarrow{e_2}-U_2V_1\,\overrightarrow{e_3}-U_3V_2\,\overrightarrow{e_1}`$
+
 
 * [ES]  <br>
 [FR] <br>
 [EN] method similar to the sum used to obtain the determinant of a matrix :<br>
 <br>$`\vec{U}\land\vec{V}=\begin{vmatrix} \overrightarrow{e_1}&\overrightarrow{e_2}&\overrightarrow{e_3}\\
 U_1 & U_2 & U_3\\V_1 & V_2 & V_3\end{vmatrix}`$
-$`=U_1V_1\overrightarrow{e_3}+U_2V_3\overrightarrow{e_1}+U_3V_1\overrightarrow{e_2}`$
-$`- U_1V_3\overrightarrow{e_2}-U_2V_1\overrightarrow{e_3}-U_3V_2\overrightarrow{e_1}`$
+$`=U_1V_1\,\overrightarrow{e_3}+U_2V_3\,\overrightarrow{e_1}+U_3V_1\,\overrightarrow{e_2}`$
+$`- U_1V_3\,\overrightarrow{e_2}-U_2V_1\,\overrightarrow{e_3}-U_3V_2\,\overrightarrow{e_1}`$