# oef vecteurs --- Introduction ---

This modulus contains at this moment 10 simple exercises on vectors and their applications in physics on the following subjects:
• Calculation of the coordinates of a vector whose norm and direction are known - Decomposition of a force following some perpendicular directions.
• Linear combination of vectors
• Applications of the scalar product of vectors (angle between vectors, work of a force)
• Vectorial product of orthogonal unit vectors : Direct or indirect coordinate system
• Numerical answers must be given using a precise number of significant figures. To familiarise your students with this notion, an exercice ("Sifnificant figures") is proposed.
• In most of the exercises, suggestions or reminders are given either by means of a direct link in the problem statement (rappel) or at the bottom of the page (indication)

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### Relation entre trois vecteurs

 xrange -200,200 yrange -200,200 arrow -50+,0+,50+,0+,10,red text red,,-10,large, arrow 0+,-50+,0+,50+,10,blue text blue,+10,,large, arrow -50+,-50+,50+,50+,10,green text green,+10,,large, xrange -200,200 yrange -200,200 arrow -50+,-50+,50+,50+,10,red text red,,-10,large, arrow 50+,-50+,-50+,50+,10,blue text blue,+10,+20,large, arrow ,-100+,,100+,10,green text green,+10,,large, xrange -200,200 yrange -200,200 arrow -50+,50+,50+,-50+,10,red text red,-30,,large, arrow -50+,-50+,50+,50+,10,blue text blue,+10,,large, arrow -100+,,100+,,10,green text green,,-10,large,

### Mobile en rotation

 Un mobile décrit un cercle de rayon R= m (représenté en jaune sur la figure), à vitesse uniforme V= m/s. A un instant donné, la position du mobile, , est représentée par le vecteur bleu , et sa vitesse, , par le vecteur rouge. Calculer les coordonnées cartésiennes de et Ry(m) = Calculer les coordonnées cartésiennes de et Vy(m) = (On donnera les réponses avec 3 chiffres significatifs). An object is moving with a circular trajectory of radius R= m (represented in yellow in the figure) with a uniform velocity V= m/s. At a given instant, the position of the object, , is represented by the blue vector , and its velocity, , by the red vector. Calculate the Cartesian coordinates of and Ry(m) = Calculate the Cartesian coordinates of and Vy(m) = (Please write the answers using 3 significant figures). xrange -1.5,1.5 yrange -1.5,1.5 arrow -1.5,0,1.5,0,10,black arrow 0,-1.5,0,1.5,10,black text black,1.4,-0.1,large,x text black,0.1,1.4,large,y text black,-0.1,-0.1,large,0 linewidth 2 circle 0,0,167,yellow fcircle cos(/180*3.1416),sin(/180*3.1416),12,black arrow 0,0,cos(/180*3.1416),sin(/180*3.1416),14,blue arrow cos(/180*3.1416),sin(/180*3.1416),1.22*cos((+35)/180*3.1416),1.22*sin((+35)/180*3.1416),14,red linewidth 1 arc 0,0,1.2,1.2,0,,blue text blue,cos(/2/180*3.1416),sin(/2/180*3.1416),large, text blue,cos(/2/180*3.1416)+0.3,sin(/2/180*3.1416)+0.05,small,O

### Angle entre deux vecteurs (<90°)

 Deux vecteurs unitaires , (représenté en ) et (représenté en ), ont pour coordonnées cartésiennes les valeurs suivantes: ax = et ay = bx = et by = A l'aide du produit scalaire, déterminer l'angle entre ces deux vecteurs. On donnera la réponse arrondie au degré le plus proche. Two unit vectors, (represented by ) et (represented by ), are defined by the following values of their Cartesian coordinates: ax = and ay = bx = and by = Use the scalar product to determine the angle between these two vectors. Please give the answer rounded to the nearest degree. = deg xrange -100,100 yrange -100,100 arrow -100,0,100,0,10,black arrow 0,-100,0,100,10,black text black,92,-10,large,x text black,10,95,large,y text black,-10,-10,large,0 arrow 0,0,80*,80*,14, arrow 0,0,80*,80*,14, arc 0,0,40,40,,,red text red,50*cos((+/2)/180*3.1416),50*sin((+/2)/180*3.1416),large,?

### Angle entre deux vecteurs (>90°)

 Deux vecteurs unitaires , (représenté en ) et (représenté en ), ont pour coordonnées cartésiennes les valeurs suivantes: ax = et ay = bx = et by = A l'aide du produit scalaire, déterminer l'angle entre ces deux vecteurs. On donnera la réponse arrondie au degré le plus proche. Two unit vectors , (represented by ) et (represented by ), are defined by the following values of their Cartesian coordinates: ax = and ay = bx = and by = Use the scalar product to determine the angle between these two vectors. Please round the answer to the nearest degree. = deg xrange -100,100 yrange -100,100 arrow -100,0,100,0,10,black arrow 0,-100,0,100,10,black text black,92,-10,large,x text black,10,95,large,y text black,-10,-10,large,0 arrow 0,0,80*,80*,14, arrow 0,0,80*,80*,14, arc 0,0,40,40,,,red text red,50*cos((+/2)/180*3.1416),50*sin((+/2)/180*3.1416),large,?

### Projection d'un vecteur

 Deux vecteurs , (représenté en bleu ) et (représenté en vert ) , ont pour coordonnées catésiennes les valeurs suivantes: Ax = cm et Ay = cm Bx = cm et By = cm Quelle est la norme du vecteur ? A= cm A l'aide du produit scalaire, déterminer la longueur du segment OH, projection de sur la direction de . OH = cm On donnera les réponses avec 3 chiffres significatifs. Two vectors , (represented in blue ) and (represented in green ) , are define by the following value of their Cartesian units: Ax = cm and Ay = cm Bx = cm and By = cm What is the norm of the vector? A= cm Use the scalar product to determine the length of th OH segment, projection of on the direction of . OH = cm Please give the answers with 3 significant figures. xrange -100,100 yrange -100,100 arrow -100,0,100,0,10,black arrow 0,-100,0,100,10,black text black,92,-10,large,x text black,10,95,large,y text black,5,-6,large,0 arrow 0,0,,,14,blue arrow 0,0,,,14,green dline ,,,,red text red,,,large,H

### Combinaison de vecteurs

 xrange -150,+150 yrange -150,+150 parallel -150,-150,150,-150,0,15,20, grey parallel -150,-150,-150,150,15,0,20, grey hline 0,0,black vline 0,0,black arrow 0,0,,,10,blue text blue,1.2*,1.2*,large,A arrow 0,0,,,10,green text green,1.2*,1.2*,large,B dline ,,,,blue dline ,,,,green arrow 0,0,,,10,red text red,1.1*,1.1*,large,C p= q=

A(,,)
• B(,,)

• (J)=

### Bloc sur un plan incliné

 m= kg N. . L= WF (kJ) = WP (kJ) = xrange -1,10 yrange -1,10 line -1,0,10,0,black line 0,0,10*cos(*3.1416/180),10*sin(*3.1416/180),black arc 0,0,3,3,0,,red text red,2*cos((/2)/180*3.1416),3*sin((/2)/180*3.1416),large, text red,0.7+2*cos((/2)/180*3.1416),0.4+3*sin((/2)/180*3.1416),small,o fpoly blue,,,,,,,, linewidth 2 arrow ,,,-3,10,black copy +0.5,-2,-1,-1,-1,-1,P.gif arrow ,,+3*sin(*3.1416/180)*cos(*3.1416/180),+3*sin(*3.1416/180)*sin(*3.1416/180),10,black copy +0.3,+1.8,-1,-1,-1,-1,F.gif linewidth 1 dline ,,,,black

### Système de coordonnées direct /indirect

xrange 0,800 yrange 0,200 --------------------------------------------------------- arrow 100-*50,100,100+*50,100,10, text ,100+*50,95,large, arrow 100,100-*50,100,100+*50,10, text ,110,100+*50,large, darrow 100-*30,100-*30,100+*30,100+*30,10, line 70,70,100,100, text ,100+*30,100+*30,large, text black, 95,25,large,(a) ------------------------------------------------------------ arrow 300-*50,100,300+*50,100,10, text ,300+*50,95,large, arrow 300,100-*50,300,100+*50,10, text ,310,100+*50,large, darrow 300-*30,100-*30,300+*30,100+*30,10, line 270,70,300,100, text ,300+*30,100+*30,large, text black, 295,25,large,(b) ------------------------------------------------------------------- arrow 500-*50,100,500+*50,100,10, text ,500+*50,95,large, arrow 500,100-*50,500,100+*50,10, text ,510,100+*50,large, darrow 500-*30,100-*30,500+*30,100+*30,10, line 470,70,500,100, text ,500+*30,100+*30,large, text black, 495,25,large,(c) ------------------------------------------------------------------- arrow 700-*50,100,700+*50,100,10, text ,700+*50,95,large, arrow 700,100-*50,700,100+*50,10, text ,710,100+*50,large, darrow 700-*30,100-*30,700+*30,100+*30,10, line 670,70,700,100, text ,700+*30,100+*30,large, text black, 695,25,large,(d)
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