Capítulo 0 Álgebra vetorial - Instituto de Matemática - UFRGS
Capítulo 0 Álgebra vetorial - Instituto de Matemática - UFRGS Capítulo 0 Álgebra vetorial - Instituto de Matemática - UFRGS
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
- Page 1 and 2: u v w u + v = v + u,
- Page 3 and 4: xyz x y z
- Page 5 and 6: u = √ 2 v = √ 5 w = √ 13 6
- Page 7 and 8: = R⊕ x = R⊕ cos φ cos λ y =
- Page 9: u · v = v · u, u · (αv + β w
- Page 13 and 14: u×v 2 det (u; v; u × v)
- Page 15: P ρ φ z z ρ
u + v ≤ (u + v) = u + v<br />
u v 90 ◦ <br />
cos (u, v) = 0 u · v = 0 ⊥ <br />
<br />
u⊥v ⇐⇒ u · v = 0 <br />
i j k <br />
i · j =i · k = j · k = 0.<br />
u =i +j v =i + 2j w = 1<br />
<br />
18, 43 ◦ 11, 3 ◦ 7, 13 ◦<br />
3 i + 1<br />
2 j <br />
α β u v <br />
<br />
cos (αu, βv) = cos (u, v) .<br />
<br />
u = u1i + u2j + u3 k u1 = u ·i u2 = u ·j u3 = u · k <br />
<br />
u = u ·i i + u · j j + u · <br />
k k.<br />
u = √ <br />
2 i + j v = 2<br />
√ <br />
2 i − j <br />
2<br />
<br />
u v<br />
− k k<br />
u = 2i +j + k v = 2i −j − k <br />
u v<br />
<br />
j − <br />
k −j + <br />
k <br />
√ 2<br />
2<br />
√ 2<br />
2<br />
u v w <br />
(u · v) w = 0 u (v · w) = 0<br />
(u · v) w = u (v · w) <br />
u =i v = j w = j u =i v =i + j w = j
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