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 dccd44038..21ad4aac7 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
@@ -709,8 +709,7 @@ $`=\dfrac{\overrightarrow{OM}(t+dt)-\overrightarrow{OM}(t)}{dt}`$<br>
 <br>
 [ES] En las expresiones anteriores, también simplificaría la escritura. Algunos ejemplos :<br>
 [FR] Sur les expressions ci-dessus, cela permettrait aussi de simplifier l'écriture. Quelques exemples : :<br>
-[EN] On the expressions above, it would also simplify the writing. Some examples :<br>
-
+[EN] On the expressions above, it would also simplify the writing. Some examples :
 
 
 * Asi / ainsi / thus :<br>
@@ -725,6 +724,16 @@ $`=d\left[A(t)\cdot\overrightarrow{e_x}\right]+d\left[B(t)\cdot\overrightarrow{e
 $`=dA(t)\cdot\overrightarrow{e_x}+A(t)\cdot d\overrightarrow{e_x}
 +d(B(t)\cdot\overrightarrow{e_y}+ B(t)\cdot d\overrightarrow{e_y}`$
 
+* Asi / ainsi / thus :<br>
+$`d\left(\overrightarrow{OM}(t)\right)=d\left(\overrightarrow{OM}(t)\right)_{||}
++d\left(\overrightarrow{OM}(t)\right)_{\perp}`$<br>
+se convertiría en / deviendrait / would become :<br>
+$`d\overrightarrow{OM}(t)=d\overrightarrow{OM}_{||}(t)
++d\overrightarrow{OM}_{\perp}(t)`$<br>
+con / avec / with <br>
+$`\overrightarrow{OM}_{||}=(\overrightarrow{OM}\cdot\overrightarrow{e_{||}})\,\overrightarrow{e_{||}}`$ and 
+$`\overrightarrow{OM}_{\perp}=(\overrightarrow{OM}\cdot\overrightarrow{e_{\perp}})\,\overrightarrow{e_{\perp}}`$ 
+