Géométrie vectorielle plane, cours, première S - MathsFG

(O;~ i;~ j)
~u ~v ~u ~v k
~v =k~u ~ 0
A B C D
•A B C ~ AB ~ AC
•(AB) (CD)~ AB ~ CD
~u ~v (x~u;y~u) (x~v ;y~v )
k ~u ~v
x~uy~v −x~v y~u = 0
•~u =k~v k x~u =kx~v y~u =ky~v x~uy~v −x~v y~u =kx~v y~v −x~v ky~v = 0
•x~uy~v −x~v y~u = 0 ~u =~ 0~v ~ 06= 0
x~u 6= 0 y~u 6= 0 x~u 6= 0 y~u
x~uy~v −x~v y~u = 0 x~uy~v =x~v y~u y~v =y~u
x~u
x~v k=x~u x~v y~v =ky~u
x~u =kx~v
A(3; 2) B(2; 5) C(−3; 8) D(2; 4)
~
AB (xB−xA;yB−yA)~ AB(5 −2; 2 −3) ~ AB(3; −1)
~
CD (5; −4)
x~uy~v −x~v y~u = 4 ×(−4) −(−3) ×5 = −1~ AB ~ CD
(AB) (CD)

(O;~ i;~ j)~u k~uk
A B ~u =~ AB
k~uk=k~ ABk=AB
•~u (x;y)
k~uk=px2+y2
•~ AB A(xA;yA)B(xB;yB)
k~ ABk=AB =p(xB−xA)2+ (yB−yA)2
A(3; 2) B(2; 5)
~
AB (−1; 3) k~ ABk=p(−1)2+ 32=√10
D~u
A B D~ AB =~u
DA ~u
DM~ AM ~u

(O;~ i;~ j)~v (1; a)
(D)y=ax +b
A(0; b)B(1; a+b)
Dy=ax +b
~
AB (1 −0; a+b−b) (1; a)
(O;~ i;~ j)
ax +by +c= 0 a b c (a;b)6= 0 a b x
y
~u (−b;a)
~u D(α;β)A
(xA;yA)
M(x;y)~ AM ~u
~
AM (x−xA;y−yA)x~uy~v −x~v y~u M
α(y−yA)−β(x−xA= 0 −βx +αy −yAα+xAβ= 0
a=−β b =α c =yAα+xAβ

(d)A(2; 3) ~u
(4; 5)
•M(x;y)~ AM ~u 4(y−3) −
5(x−2) = 0 4y−12 −5x+ 10 = 0 −5x+ 4y−2=0
•~u(4; 5) −b= 4 a= 5 a= 5 b=−4
5x−4y+c= 0 A(2; 3)
5×2−4×3 + c= 0 −2 + c= 0 c= 2 5x−4y+ 2 = 0
ax+by +c= 0 a b c
(a;b)6= (0; 0) ~u
(−b;a)
b6= 0 y=−ax
b−c b
(1; −a
b) (−b;a)b
b= 0 ax =−c x =−c
aa6= 0
(0; a) (−b;a)
ax +by +c= 0 (a;b)6= (0; 0)
3x+ 2y−5=0 ~u(−2; 3)