Capítulo 0 Álgebra vetorial - Instituto de Matemática - UFRGS

u v w
u + v = v + u,
u + (v + w) = (v + u) + w,
(α + β) · u = α · u + β · v,
α · (u + v) = α · u + α · v,
α · (βu) = (αβ) · u
0 + v = v,
0 · v = 0,
1 · v = v.
u +v + w

0 0
u − v = u + (−1) · v.
(−1) · v −v
v + (−v) = v + (−1)v = (1 − 1)v = 0v = 0.
α · v αv
{v1, v2, . . . , vn}
{α1, α2, . . . , αn} αi = 0
n
αivi = 0
i=1
{v1, v2, . . . , vn}
n
αivi = 0
i=1
α1 = α2 = . . . = αn = 0.
E = {e1, e2, . . . , en}
V v ∈ V B
n
v = αiei.
i=1
V E = {e1, e2, . . . , en} F = { f1, f2, . . . , fm}
V n = m
B1 B2
B1 B2
n R n

xyz
x y z
u v w
det (u; v; w)
u v w
i j k
v = 〈v1, v2, v3〉 .
i = 〈1, 0, 0〉
j = 〈0, 1, 0〉
k = 〈0, 0, 1〉
v = v1 i + v2 j + v3 k.
0 = 0i + 0j + 0 k = 〈0, 0, 0〉 .
u = u1 i +
u2 j + u3 k v = v1 i + v2 j + v3 k
u + v = (u1 + v1)i + (u2 + v2)j + (u3 + v3) k.
u = u1 ·i + u2 · j + u3 · k
αu = (αu1)i + (αu2)j + (αu3) k.

v
v
v = v2 1 + v2 2 + v2 3.
αu = |α| u,
u + v ≤ u + v,
u = 0 =⇒ u = 0,
v
v
v = v
v = vˆv
v v ˆv
ˆv = v
.
v
ˆv v ˆv
v
i j k
u =i+j v =i+2j w = 1i+
1
3 2 j
û ˆv ˆw

u = √ 2 v = √ 5 w = √ 13
6 û = √ 2
2 i+ √ 2
2 j ˆv = √ 5
5 i+ 2√ 5
5 j ˆw = 2√ 13
13 i+ 3√ 13
13 j
u = cos ϕi + sen ϕj
ϕ = π
ϕ = 0 ϕ = π π ϕ = 6 2
u = sen θ cos ϕi+sen θ sen ϕj +cos θ k
θ = 0
θ = π
4
θ = π
2
θ = π
ϕ = π
4
ϕ = π
4
u = u1 i+u2 j xy v = v
cos ϕi + sen ϕj
xy m(ϕ) = u + v
m(ϕ)
u v w
u1
det (u; v; w) = u2
v1
v2
w1
w2
> 0.
u3 v3 w3
u = u1 i + u2 j + u3 k v = v1 i + v2 j + v3 k w = w1 i + w2 j + w3 k
u v w v u w
u v w v w u w u v
u v w u =i +j v = −2i +j w =i +j + k
u v w u = i + j v = −2i + j w = i
•
• z
• xz x > 0
R⊕ λ
φ
r = xi+yj +z k

x y z r λ φ
d
δ
δ = R⊕ cos −1
1 − d2
2R2
⊕
d = 0 d = 2R⊕
R⊕ = 6378Km
λ φ
xyz
30 ◦ 01 ′ 58 ′′ 51 ◦ 13 ′ 48 ′′
51 ◦ 30 ′ 28 ′′ 0 ◦ 7 ′ 41 ′′
35 ◦ 41 ′ 22 ′′ 139 ◦ 41 ′ 30 ′′

= R⊕ x = R⊕ cos φ cos λ y = R⊕ cos φ sen λ z = R⊕ sen φ
φ λ x y z
−30, 0328 ◦ −51, 23 ◦ 3457, 65 −4305, 07 −3192, 16
51, 5078 ◦ −0, 0781 ◦
35, 6894 ◦ 139, 6917 ◦
λ φ
48 ◦ 51 ′ 30 ′′ 0 ◦ 02 ′ 24 ′′
27 ◦ 10 ′ 27 ′′ 0 ◦ 58 ′ 42 ′′
51 ◦ 10 ′ 44 ′′ 0 ◦ 01 ′ 55 ′′

u
v < u, v > u · v
< u, v >= u · v = u1v1 + u2v2 + u3v3
u v w
w = u − v.
w 2 = u 2 + v 2 − 2uv cos θ
u = 0 v = 0
cos θ = w2 − u2 − v2 .
2uv
u 2 = u 2 1 + u 2 2 + u 2 3
v 2 = v 2 1 + v 2 2 + v 2 3
w 2 = w 2 1 + w 2 2 + w 2 3 = (u1 − v1) 2 + (u2 − v2) 2 + (u3 − v3) 2
cos θ = w2 − u 2 − v 2
2uv
= u1v1 + u2v2 + u3v3
uv
= u · v
uv
u · v = uv cos (u, v)
cos (u, v) u v
0

u · v = v · u,
u · (αv + β w) = α(u · v) + β(u · w),
u · u = u 2 ,
|u · v| ≤ uv,
α(u·v) = (αu)·v αu·v
(αu + βv) · w = α(u · w) + β(v · w)
−1 ≤ cos θ ≤ 1.
û ˆv
û + ˆv 2 = (û + ˆv) · (û + ˆv) = 2 + 2û · ˆv
û − ˆv 2 = (û − ˆv) · (û − ˆv) = 2 − 2û · ˆv
û · û = ˆv · ˆv = 1
û + ˆv 2 ≥ 0 û − ˆv 2 ≥ 0
−1 ≤ û · ˆv ≤ 1
|û · ˆv| ≤ 1 u = uû v = vˆv
|u · v| ≤ uv
u + v 2 = (u + v) · (u + v) = u 2 + 2u · v + v 2
u · v ≤ |u · v| ≤ uv
u + v 2 ≤ u 2 + 2uv + v 2 = (u + v) 2

u + v ≤ (u + v) = u + v
u v 90 ◦
cos (u, v) = 0 u · v = 0 ⊥
u⊥v ⇐⇒ u · v = 0
i j k
i · j =i · k = j · k = 0.
u =i +j v =i + 2j w = 1
18, 43 ◦ 11, 3 ◦ 7, 13 ◦
3 i + 1
2 j
α β u v
cos (αu, βv) = cos (u, v) .
u = u1i + u2j + u3 k u1 = u ·i u2 = u ·j u3 = u · k
u = u ·i i + u · j j + u ·
k k.
u = √
2 i + j v = 2
√
2 i − j
2
u v
− k k
u = 2i +j + k v = 2i −j − k
u v
j −
k −j +
k
√ 2
2
√ 2
2
u v w
(u · v) w = 0 u (v · w) = 0
(u · v) w = u (v · w)
u =i v = j w = j u =i v =i + j w = j

u = cos(θ1)i + sen(θ1)j v = cos(θ2)i + sen(θ2)j
cos (u, v) = cos(θ1 − θ2).
θ u v
|θ1 − θ2|, |θ1 − θ2| ≤ 180
θ =
◦
360 ◦ − |θ1 − θ2|, |θ1 − θ2| > 180 ◦
θ1 θ2 0 360 ◦
u v
u + v û = ˆv û = −ˆv
u + v 2 = u 2 + v 2 + 2u · v
u = u1 i + u2 j + u3 k v = v1 i + v2 j + v3 k u ×v
u × v = (u2v3 − u3v2)i + (u3v1 − u1v3)j + (u1v2 − u2v1) k
u × v = −v × u,
(αu + βv) × w = α (u × w) + β (v × w) ,
u × (αv + β w) = α (u × v) + β (u × w) ,
(u × v) · u = (u × v) · v = 0,
u × v = uv sen(u, v).
det (u; v; u × v) = u 2 v 2 sen 2 (u, v) > 0.
sen(u, v) u v
u v
u × v = 0 det (u; v; u × v) = 0
u v u × v

(u × v) · u = (u2v3 − u3v2)i + (u3v1 − u1v3)j + (u1v2 − u2v1)
k u1i + u2j + u3
k
(u × v) · v =
= (u2v3 − u3v2) u1 + (u3v1 − u1v3) u2 + (u1v2 − u2v1) u3 = 0
(u2v3 − u3v2)i + (u3v1 − u1v3)j + (u1v2 − u2v1)
k v1i + v2j + v3
k
= (u2v3 − u3v2) v1 + (u3v1 − u1v3) v2 + (u1v2 − u2v1) v3 = 0
u × v 2 + |u · v| 2 = u 2 v 2
u × v 2 =
(u2v3 − u3v2)i + (u3v1 − u1v3)j + (u1v2 − u2v1)
k
= (u2v3 − u3v2) 2 + (u3v1 − u1v3) 2 + (u1v2 − u2v1) 2
= u 2 2v 2 3 − 2u2u3v2v3 + u 2 3v 2 2
2 + u3v 2 1 − 2u1u3v1v3 + u 2 1v 2 3
+ u 2 1v 2 2 − 2u1u2v1v2 + u 2 2v 2 1 .
|u · v| 2 = (u1v1 + u2v2 + u3v3) 2 = u 2 1v 2 1 + u 2 2v 2 2 + u 2 3v 2 3 + 2u1u2v1v2 + 2u1u3v1v3 + 2u2u3v2v3.
u × v 2 + |u · v| 2 = (u1v1 + u2v2 + u3v3) 2
= u 2 1v 2 1 + u 2 1v 2 2 + u 2 1v 2 3 + u 2 2v 2 1 + u 2 2v 2 2 + u 2 2v 2 3 + u 2 3v 2 1 + u 2 3v 2 2 + u 2 3v 2 3
= u 2 1 + u 2 2 + u 2 2
3 v1 + v 2 2 + v 2 2 2
3 = u v .
u × v 2 = u 2 v 2 − |u · v| 2 = u 2 v 2 − [uv cos (u, v)] 2 = u 2 v 2 1 − cos 2 (u, v) = u 2 v 2 sen 2 (u, v)
sen (u, v) ≥ 0
u v
sen 0 = sen 180◦ = 0 0
u1 v1 (u2v3
− u3v2)
det (u; v; u × v) = u2 v2 (u3v1
− u1v3)
u3 v3 (u1v2 − u2v1)
= u 2 1v 2 2
2 − u1u2v1v2 + u3v 2 2
1 − u1u3v1v3 + u2v 2
3 − u2u3v2v3
− u1u2v1v2 − u 2 2v 2
1 − u1u3v1v3 − u 2 1v 2
3 − u2u3v2v3 − u 2 3v 2 2
2

u×v 2
det (u; v; u × v)
w = u × v
u v w
w u
v
u v
u × v = uv sen (u, v) ê
ê u v
u v
u v
u v
i j
k
u × v =
u1 u2 u3
v1 v2 v3

i j k
i ×i = 0, i × j = k, i × k = −j
j ×i = − k, j × j = 0, j × k =i
k ×i = j, k × j = −i, k × k = 0
u =i + 2j v = 3i − 2j w = u × v
i
w =
u1
j
u2
k
u3
=
i
1
3
j
2
−2
k
0
0 =i(0 − 0) + j(0 − 0) + k(−2 − 6) = −8k v1 v2 v3
w =
u × v = i + 2j × 3i − 2j = 3(i ×i) − 2(i × j) + 6(j ×i) − 4(j × j)
= 30 − 2 k − 6 k − 40 = −8 k
u v w u × (v × w) = (u × v) × w
u =i v =i w = k
u × u
u × û
u · u
u · û
(u + v) · (u + v)
(u + v) × (u + v)
(u − v) · (u − v)
(u − v) × (u − v)
(u + v) · (u − v)
(u + v) × (u − v)
00u 2 uu 2 + 2u · v + v 2 0 u 2 − 2u · v + v 2 0 u 2 − v 2 2v × u
u · (v × w) = det (u; v; w) u v
w u · (v × w) > 0 u · (v × w) < 0

P ρ
φ z z
ρ
Q P xy φ
x > 0
x = ρ cos φ
y = ρ sen φ
ρ = x 2 + y 2
ρ φ z
cos φ = x
ρ =
sen φ = y
ρ =
x
x 2 + y 2
y
x 2 + y 2
〈1, 1, 1〉
〈1, −1, 1〉
〈−1, 1, 1〉
〈−1, −1, 1〉
√ 2, π
4 , 1 √ 2, 5π
4 , 1 √ 2, 3π
4 , 1 √ 2, 7π
4 , 1
ρ 2 + z 2