DÉTERMINANTS DANS LE PLAN ET DANS L`ESPACE

R2R3
~u ·~v k~uk
~u = (x1, y1)~v = (x2, y2)R2
~u ~v
{α~u +β~v |α, β ∈[0,1]}
~u
~v
~u +~v

A
A=b×h
h
b
b
h
A ~u ~v
~u0= (−y1, x1)
~u
|~v ·~u0|=h× k~uk=hb =A
~v
h
b
~u
~u0
A=|x1y2−y1x2|
a, b, c, d
a c
b d
:= ad −bc

B= (~ı,~)~u =x1 ~ı +y1~
~v =x2 ~ı +y2~ R2
(~u, ~v)B
DetB(~u, ~v) := 
x1x2
y1y2
=x1y2−y1x2
~u =x1 ~ı +y1~ ~u = (x1, y1)
B
DetB(~ı,~ı +~) = 
1 1
0 1
= 1 ×1−0×1 = 1
DetB(~ı + 2~, 3~ı + 4~) = 
1 3
2 4
= 1 ×4−2×3 = −2
~u, ~v, ~s R2λ∈R
DetB(~v, ~u) = −DetB(~u, ~v)
DetB(~u, ~u) = 0
DetB(λ~u, ~v) = λDetB(~u, ~v) DetB(~u +~s, ~v) =
DetB(~u, ~v) + DetB(~s, ~v)
DetB(~u, λ~v) = λDetB(~u, ~v) DetB(~u, ~v +~s) =
DetB(~u, ~v) + DetB(~u, ~s)

~s
~v +~s
~u
~v
λ~u
~v
(x1, y1)~u B
(x2, y2)~v (a, b)~s
DetB(~u, ~v) = 
x1x2
y1y2
=x1y2−y1x2
DetB(~v, ~u) = 
x2x1
y2y1
=x2y1−y2x1=−(x1y2−y1x2)
DetB(~u, ~u) = −DetB(~u, ~u)
DetB(λ~u, ~v) = (λx1)y2−(λy1)x2
=λ(x1y2−y1x2)
=λDetB(~u, ~v)
DetB(~u +~s, ~v) = (x1+a)y2−(y1+b)x2
= (x1y2−y1x2)+(ay2−bx2)
= DetB(~u, ~v) + DetB(~s, ~v)

DetB(~u, λ~v) = −DetB(λ~v, ~u)
=−λDetB(~v, ~u)
=λDetB(~u, ~v)
DetB(~u, ~v +~s) = −DetB(~v +~s, ~u)
=−DetB(~v, ~u)−DetB(~s, ~u)
= DetB(~u, ~v) + DetB(~u, ~s)
~u, ~v R2λ∈R
DetB(~u, ~v +λ~u) = DetB(~u, ~v)
DetB(~u, ~v +λ~u) = DetB(~u, ~v) + λDetB(~u, ~u)
= DetB(~u, ~v)+0
~u
~v
~v +λ~u
h
b
h b
3
~u ~v DetB(~u, ~v) = 0