Bonjour
Exercice 1 :
1) u.v = xu*xv + yu*yv = 4*(-1) + 3*1 = -4 + 3 = -1
2) ll u ll = √(xu² + yu²) = √(4² + 3²) = √25 = 5
ll v ll = ✓(xv² + yv²) = √((-1)² + 1²) = √2
3) u.v = ll u ll * ll v ll * cos(u ; v) = 5√2 cos(u ; v)
5√2 cos(u ; v) = -1
cos(u ; v) = -1/5√2
(u ; v) = arccos(-1/(5√2)) ≈ 98°
Exercice 2 :
1) ll u ll = √(xu² + yu²) = √((-2)² + 5²) = √29
2) u.v = ll u ll * ll v ll * cos(u ; v) = 6√29 cos(u ; v)
6√29 cos(u ; v) = 12
cos(u ; v) = 2/√29
(u ; v) = arccos(2/√29) ≈ 68°
Exercice 3 :
1) AB (xB - xA ; yB - yA) AB (-14 ; 26)
AC (xC - xA ; yC - yA) AC (16 ; -5)
2) ll AB ll = √(xAB² + yAB²) = √((-14)² + 26²) = 2√218
ll AC ll = √(xAC² + yAC²) = √(16² + (-5)²) = √281
3) AB.AC = xAB * xAC + yAB * yAC = -14*16 + 26*(-5)
AB.AC = -354
4) AB.AC = ll AB ll * ll AC ll * cos(AB ; AC)
AB.AC = 2√218 √281 cos(^BAC)
2√61258 cos(^BAC) = -354
cos(^BAC) = -177/√61258
^BAC = arccos(-177/√61258) ≈ 135,7°
