From 282dfbbc9f1398817f8de64958c18c8545819603 Mon Sep 17 00:00:00 2001
From: Claude Meny <claude.meny@irap.omp.eu>
Date: Wed, 19 Aug 2020 17:53:17 +0200
Subject: [PATCH] Update textbook.fr.md

---
 .../vector-analysis/textbook.fr.md            | 24 ++++++++-----------
 1 file changed, 10 insertions(+), 14 deletions(-)

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 a3cdf02c0..5fc3027fe 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
@@ -407,22 +407,18 @@ $`\;\Longrightarrow\left|\begin{array}{l}\overrightarrow{U}\cdot\overrightarrow{
 
 ##### Producto escalar de dos vectores ortogonales / Produit scalaire de 2 vecteurs orthogonaux / 
 
-
-$`\forall \overrightarrow{U}\in\mathcal{P}\quad\overrightarrow{U}=U_a\cdot\overrightarrow{a}+U_b\cdot\overrightarrow{b}`$<br>
-$`\forall \overrightarrow{V}\in\mathcal{P}\quad\overrightarrow{V}=V_a\cdot\overrightarrow{a}+V_b\cdot\overrightarrow{b}`$<br>
-$`\overrightarrow{U}\cdot\overrightarrow{V}=(U_a\cdot\overrightarrow{a}+U_b\cdot\overrightarrow{b})\cdot (V_a\cdot\overrightarrow{a}+V_b\cdot\overrightarrow{b})`$<br>
-$` = U_a\,V_a\,(\overrightarrow{a}\cdot\overrightarrow{a})+U_a\,V_b\,(\overrightarrow{a}\cdot \overrightarrow{b})`$
-$`+U_b\,V_a\,(\overrightarrow{b}\cdot \overrightarrow{a})+U_b\,V_b\,(\overrightarrow{b}\cdot\overrightarrow{b})`$<br>
-$`= U_a\,V_a\,\overrightarrow{a}^2 + U_b\,V_b\,\overrightarrow{b}^2 + (U_a\,V_a+U_b\,V_a)\,(\overrightarrow{a}\cdot \overrightarrow{b})`$
-
+$`\forall \overrightarrow{U}\in\mathcal{P}\quad, \forall \overrightarrow{V}\in\mathcal{P}`$
+$`\overrightarrow{U}\perp\overrightarrow{V}\Longleftrightarrow\widehat{\overrightarrow{U},
+\overrightarrow{V}}=\dfrac{\pi}{2}\Longleftrightarrow cos(\widehat{\overrightarrow{U},
+\overrightarrow{V}})=O`$**$`\overrightarrow{U}\cdot\overrightarrow{V}=0`$**.
 
 ##### Producto escalar de dos vectores en una base ortonormal del espacio / Prduit scalaire de deux vecteurs 2 vecteurs dans une base orthonormée de l'espace / Scalar product of 2 vectors in an orthonormal basis
 
 "$`(\vec{e_1},\vec{e_2},...,\vec{e_n})`$ est une base de $`\mathcal{E}`$"
 $`\quad\Longrightarrow`$
 $`\displaystyle\quad\forall \overrightarrow{U}\in\mathcal{P}\quad \overrightarrow{U}=\sum_{i=1}^n\;U_i\cdot\vec{e_i}`$ 
-$`\displaystyle\quad\forall \overrightarrow{V}\in\mathcal{P}\quad \overrightarrow{U}=\sum_{i=1}^n\;V_i\cdot\vec{e_i}`$  
-$`\displaystyle\quad\overrightarrow{U}\cdot\overrightarrow{V}=U_1\,V_1 + U_2\,V_2 + ... + U_n\,V_n = \sum_{i=1}^n\;U_i\,V_i`$
+$`\displaystyle\quad\forall \overrightarrow{V}\in\mathcal{P}\quad \overrightarrow{V}=\sum_{i=1}^n\;V_i\cdot\vec{e_i}`$  
+**$`\displaystyle\quad\overrightarrow{U}\cdot\overrightarrow{V}=U_1\,V_1 + U_2\,V_2 + ... + U_n\,V_n = \sum_{i=1}^n\;U_i\,V_i`$**
 
 
 ##### Cálculo del ángulo entre 2 vectores en una base ortonormal del espacio / Calcul de l’angle entre 2 vecteurs dans une base orthonormée de l'espace / Calculation of the angle between 2 vectors in an orthonormal basis
@@ -436,10 +432,10 @@ $`\quad\Longrightarrow\quad cos (\widehat{\overrightarrow{U},\overrightarrow{V}}
 {||\overrightarrow{U}||\cdot||\overrightarrow{V}||}`$
 $`\quad\Longrightarrow\quad cos (\widehat{\overrightarrow{U},\overrightarrow{V}})=\dfrac{U_1\,V_1 + U_2\,V_2 + U_3\,V_3}
 {||\overrightarrow{U}||\cdot||\overrightarrow{V}||}`$
-$`\quad\Longrightarrow\quad \widehat{\overrightarrow{U},\overrightarrow{V}}= arcos\left(\dfrac{\overrightarrow{U}\cdot\overrightarrow{V}}
-{||\overrightarrow{U}||\cdot||\overrightarrow{V}||}\right)`$
-$`\quad\Longrightarrow\quad \widehat{\overrightarrow{U},\overrightarrow{V}}= arcos\left(\dfrac{U_1\,V_1 + U_2\,V_2 + U_3\,V_3}
-{||\overrightarrow{U}||\cdot||\overrightarrow{V}||}\right)`$
+**$`\quad\Longrightarrow\quad \widehat{\overrightarrow{U},\overrightarrow{V}}= arcos\left(\dfrac{\overrightarrow{U}\cdot\overrightarrow{V}}
+{||\overrightarrow{U}||\cdot||\overrightarrow{V}||}\right)`$**
+**$`\quad\Longrightarrow\quad \widehat{\overrightarrow{U},\overrightarrow{V}}= arcos\left(\dfrac{U_1\,V_1 + U_2\,V_2 + U_3\,V_3}
+{||\overrightarrow{U}||\cdot||\overrightarrow{V}||}\right)`$**
 
 #### Produit vectoriel de 2 vecteurs