Formulas geral para geometria analitica

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RESUMO DE FÓRMULAS PARA GEOMETRÍA ANALÍTICA B: 1ª ÁREA
Prof. Elismar R. Oliveira

Soma de vetores:
Soma dos vetores=> Regra do Paralelogramo =>
Em coordenadas: &#3627408534; =(&#3627408537;
&#3627409359;,&#3627408538;
&#3627409359;,&#3627408539;
&#3627409359;) e &#3627408535; =(&#3627408537;
&#3627409360;,&#3627408538;
&#3627409360;,&#3627408539;
&#3627409360;)

&#3627408534; + &#3627408535; =(&#3627408537;
&#3627409359;+&#3627408537;
&#3627409360;,&#3627408538;
&#3627409359;+&#3627408538;
&#3627409360;,&#3627408539;
&#3627409359;+&#3627408539;
&#3627409360;)

Multiplicação p/ nº real
Em coordenadas
&#3627408534; =(&#3627408537;
&#3627409359;,&#3627408538;
&#3627409359;,&#3627408539;
&#3627409359;) &#3627409206;, um número real. &#3627409206;&#3627408534; =(&#3627409206;&#3627408537;
&#3627409359; ,&#3627409206; &#3627408538;
&#3627409359;,&#3627409206;&#3627408539;
&#3627409359;)
LD ou LI
1 vetor

{&#3627408534; }=
??????&#3627408491;,&#3627408532;&#3627408518; &#3627408534; =&#3627409358;
????????????,&#3627408516;&#3627408514;&#3627408532;&#3627408528; &#3627408516;&#3627408528;&#3627408527;&#3627408533;&#3627408531;á&#3627408531;&#3627408522;&#3627408528;

2 vetores

{&#3627408534; ,&#3627408535; }=
??????&#3627408491;,&#3627408532;&#3627408518; &#3627408534; =&#3627409206;&#3627408535; &#3627408528;&#3627408534; &#3627408535; =&#3627409206;&#3627408534;
????????????,&#3627408516;&#3627408514;&#3627408532;&#3627408528; &#3627408516;&#3627408528;&#3627408527;&#3627408533;&#3627408531;á&#3627408531;&#3627408522;&#3627408528;

3 vetores (em coordenadas)
Se &#3627408534; =(&#3627408537;
&#3627409359;,&#3627408538;
&#3627409359;,&#3627408539;
&#3627409359;), &#3627408535; =(&#3627408537;
&#3627409360;,&#3627408538;
&#3627409360;,&#3627408539;
&#3627409360;) e
&#3627408536; =(&#3627408537;
&#3627409361;,&#3627408538;
&#3627409361;,&#3627408539;
&#3627409361;),
&#3627408517;=&#3627408517;&#3627408518;&#3627408533;
&#3627408537;
&#3627409359;&#3627408538;
&#3627409359;&#3627408539;
&#3627409359;
&#3627408537;
&#3627409360;&#3627408538;
&#3627409360;&#3627408539;
&#3627409360;
&#3627408537;
&#3627409361;&#3627408538;
&#3627409361;&#3627408539;
&#3627409361;

{&#3627408534; ,&#3627408535; ,&#3627408536; }=
??????&#3627408491;,&#3627408532;&#3627408518; &#3627408517;=&#3627409358;
????????????,&#3627408532;&#3627408518; &#3627408517;≠&#3627409358;

Norma em coordenadas &#3627408534; =(&#3627408537;,&#3627408538;,&#3627408539;)

| &#3627408534; | = &#3627408537;
&#3627409360;
+&#3627408538;
&#3627409360;
+&#3627408539;
&#3627409360;

Produto interno

&#3627408534; ⋅&#3627408535; =
&#3627409358;,&#3627408532;&#3627408518; &#3627408534; =&#3627409358; &#3627408528;&#3627408534; &#3627408535; =&#3627409358;
| &#3627408534; | &#3627408535; &#3627408516;&#3627408528;&#3627408532; ??????,
&#3627408516;&#3627408514;&#3627408532;&#3627408528; &#3627408516;&#3627408528;&#3627408527;&#3627408533;&#3627408531;á&#3627408531;&#3627408522;&#3627408528;

Em coordenadas:
&#3627408534; ⋅&#3627408535; =&#3627408537;
&#3627409359;&#3627408537;
&#3627409360;+ &#3627408538;
&#3627409359;&#3627408538;
&#3627409360;+&#3627408539;
&#3627409359;&#3627408539;
&#3627409360;
&#3627408534; ⋅&#3627408534; =| &#3627408534;

&#3627409360;

Ou | &#3627408534; | = &#3627408534; ⋅&#3627408534;
&#3627408534; ⋅&#3627408535; =&#3627409358;<=> &#3627408534; ⊥&#3627408535;
Ângulo e Projeção &#3627408534; = &#3627408537;
&#3627409359;,&#3627408538;
&#3627409359;,&#3627408539;
&#3627409359;
e &#3627408535; =(&#3627408537;
&#3627409360;,&#3627408538;
&#3627409360;,&#3627408539;
&#3627409360;)

Ângulo ??????=∢(&#3627408534; ,&#3627408535; ): &#3627408516;&#3627408528;&#3627408532; ??????=
&#3627408534; ⋅&#3627408535;
| &#3627408534; | &#3627408535;

Projeção ortogonal de &#3627408534; sobre &#3627408535; :
&#3627408529;&#3627408531;&#3627408528;&#3627408523;
&#3627408535; &#3627408534; =
&#3627408534; ⋅&#3627408535;
&#3627408535; ⋅&#3627408535;
&#3627408535;
Produto Vetorial

&#3627408534; ∧&#3627408535; = ||
&#3627409358;,&#3627408532;&#3627408518; {&#3627408534; ,&#3627408535;} é ??????&#3627408491;
&#3627408534; ∧&#3627408535; ||=| &#3627408534; | &#3627408535; &#3627408532;&#3627408518;&#3627408527; ??????,
&#3627408516;&#3627408514;&#3627408532;&#3627408528; &#3627408516;&#3627408528;&#3627408527;&#3627408533;&#3627408531;á&#3627408531;&#3627408522;&#3627408528;

Em coordenadas:

&#3627408534; ∧&#3627408535; =&#3627408517;&#3627408518;&#3627408533;
&#3627408522; &#3627408523; &#3627408524;
&#3627408537;
&#3627409359;&#3627408538;
&#3627409359;&#3627408539;
&#3627409359;
&#3627408537;
&#3627409360;&#3627408538;
&#3627409360;&#3627408539;
&#3627409360;

&#3627408534; ∧&#3627408535; ⊥&#3627408534; e &#3627408534; ∧&#3627408535; ⊥&#3627408535;

||&#3627408534; ∧&#3627408535; ||= Área do paralelogramo
formado por &#3627408534; e &#3627408535; .

Produto Misto

&#3627408534; ,&#3627408535; ,&#3627408536; =&#3627408534; ∧&#3627408535; ⋅&#3627408536;

Em coordenadas:
&#3627408534; ,&#3627408535; ,&#3627408536; =&#3627408517;&#3627408518;&#3627408533;
&#3627408537;
&#3627409359;&#3627408538;
&#3627409359;&#3627408539;
&#3627409359;
&#3627408537;
&#3627409360;&#3627408538;
&#3627409360;&#3627408539;
&#3627409360;
&#3627408537;
&#3627409361;&#3627408538;
&#3627409361;&#3627408539;
&#3627409361;

| &#3627408534; ,&#3627408535; ,&#3627408536; |= Volume do
paralelepípedo formado por &#3627408534; ,&#3627408535; e &#3627408536; .

Vetor entre A e B
&#3627408488;&#3627408489; =&#3627408489;−&#3627408488;= &#3627408537;
&#3627409360;−&#3627408537;
&#3627409359;,&#3627408538;
&#3627409360;−&#3627408538;
&#3627409359;,&#3627408539;
&#3627409360;−&#3627408539;
&#3627409359;
onde, &#3627408488;= &#3627408537;
&#3627409359;,&#3627408538;
&#3627409359;,&#3627408539;
&#3627409359;
e &#3627408489;= &#3627408537;
&#3627409360;,&#3627408538;
&#3627409360;,&#3627408539;
&#3627409360;

Mudanças de
coordenadas
??????
&#3627409359;={&#3627408502;,&#3627408518;
&#3627409359; ,&#3627408518;
&#3627409360; ,&#3627408518;
&#3627409361; }=> ??????
&#3627409360;={&#3627408502;′,&#3627408518;
&#3627409359; ,&#3627408518;
&#3627409360; ,&#3627408518;
&#3627409361; }

??????= &#3627408537;,&#3627408538;,&#3627408539;
??????&#3627409359;

&#3627408502;′= &#3627408521;,&#3627408524;,&#3627408526;
??????&#3627409359;

??????= &#3627408537;′,&#3627408538;′,&#3627408539;′
??????&#3627409360;

&#3627408537;=&#3627408537;
′
+ &#3627408521;
&#3627408538;=&#3627408538;
′
+&#3627408524;
&#3627408539;=&#3627408539;
′
+&#3627408526;

??????
&#3627409359;={&#3627408502;,&#3627408518;
&#3627409359; ,&#3627408518;
&#3627409360; ,&#3627408518;
&#3627409361; }=> ??????
&#3627409360;={&#3627408502;,&#3627408519;
&#3627409359;
,&#3627408519;
&#3627409360;
,&#3627408519;
&#3627409361;
}

??????= &#3627408537;,&#3627408538;,&#3627408539;
??????&#3627409359;
e ??????= &#3627408537;′,&#3627408538;′,&#3627408539;′
??????&#3627409360;

&#3627408519;
&#3627409359;
=&#3627408514;
&#3627409359;&#3627409359;&#3627408518;
&#3627409359; +&#3627408514;
&#3627409360;&#3627409359; &#3627408518;
&#3627409360; +&#3627408514;
&#3627409361;&#3627409359; &#3627408518;
&#3627409361;
&#3627408519;
&#3627409360;
=&#3627408514;
&#3627409359;&#3627409360;&#3627408518;
&#3627409359; +&#3627408514;
&#3627409360;&#3627409360; &#3627408518;
&#3627409360; +&#3627408514;
&#3627409361;&#3627409360; &#3627408518;
&#3627409361;
&#3627408519;
&#3627409361;
=&#3627408514;
&#3627409359;&#3627409361;&#3627408518;
&#3627409359; +&#3627408514;
&#3627409360;&#3627409361; &#3627408518;
&#3627409360; +&#3627408514;
&#3627409361;&#3627409361; &#3627408518;
&#3627409361;

&#3627408537;=&#3627408514;
&#3627409359;&#3627409359;&#3627408537;′+&#3627408514;
&#3627409359;&#3627409360;&#3627408538;′+&#3627408514;
&#3627409359;&#3627409361;&#3627408539;′
&#3627408538;=&#3627408514;
&#3627409360;&#3627409359;&#3627408537;′+&#3627408514;
&#3627409360;&#3627409360;&#3627408538;′+&#3627408514;
&#3627409360;&#3627409361; &#3627408539;′
&#3627408539;=&#3627408514;
&#3627409361;&#3627409359;&#3627408537;′+&#3627408514;
&#3627409361;&#3627409360; &#3627408538;′+&#3627408514;
&#3627409361;&#3627409361; &#3627408539;′

Equações da reta
Vetorial
&#3627408531;: ??????=&#3627408488;+?????? &#3627408535;
&#3627408488;=(&#3627408537;
&#3627409358;,&#3627408538;
&#3627409358;,&#3627408539;
&#3627409358;) e &#3627408535; = &#3627408514;,&#3627408515;,&#3627408516; .
Paramétrica
&#3627408531;:
&#3627408537;=&#3627408537;
&#3627409358;+?????? &#3627408514;
&#3627408538;=&#3627408538;
&#3627409358;+?????? &#3627408515;
&#3627408539;=&#3627408539;
&#3627409358;+?????? &#3627408516;

Simétrica
&#3627408537;−&#3627408537;
&#3627409358;
&#3627408514;
=
&#3627408538;−&#3627408538;
&#3627409358;
&#3627408515;
=
&#3627408539;−&#3627408539;
&#3627409358;
&#3627408516;

Ângulo entre retas
&#3627408531;: ??????=&#3627408488;+?????? &#3627408534;
&#3627408532;: ??????=&#3627408489;+?????? &#3627408535;
Ângulo:
??????=∢(&#3627408531;,&#3627408532;)
&#3627408516;&#3627408528;&#3627408532; ??????=
|&#3627408534; ⋅&#3627408535; |
| &#3627408534; | &#3627408535;

Posição relativa entre as
retas r e s:

&#3627408531;: ??????=&#3627408488;+?????? &#3627408534;
&#3627408532;: ??????=&#3627408489;+&#3627409207; &#3627408535;

Paralelismo

&#3627408531; ∥&#3627408532; <=>&#3627408534; ∥&#3627408535;

Concorrentes X Reversas
&#3627408534; =(&#3627408514;
&#3627409359;,&#3627408515;
&#3627409359;,&#3627408516;
&#3627409359;), &#3627408535; =(&#3627408514;
&#3627409360;,&#3627408515;
&#3627409360;,&#3627408516;
&#3627409360;) e &#3627408488;&#3627408489; =&#3627408489;−&#3627408488;= &#3627408537;
&#3627409360;−&#3627408537;
&#3627409359;,&#3627408538;
&#3627409360;−&#3627408538;
&#3627409359;,&#3627408539;
&#3627409360;−&#3627408539;
&#3627409359;
,

&#3627408517;=&#3627408517;&#3627408518;&#3627408533;
&#3627408514;
&#3627409359;&#3627408515;
&#3627409359;&#3627408516;
&#3627409359;
&#3627408514;
&#3627409360;&#3627408515;
&#3627409360;&#3627408516;
&#3627409360;
&#3627408537;
&#3627409360;−&#3627408537;
&#3627409359;&#3627408538;
&#3627409360;−&#3627408538;
&#3627409359;&#3627408539;
&#3627409360;−&#3627408539;
&#3627409359;
=
&#3627408490;&#3627408528;&#3627408527;&#3627408516;&#3627408528;&#3627408531;&#3627408531;&#3627408518;&#3627408527;&#3627408533;&#3627408518;&#3627408532;, &#3627408532;&#3627408518; &#3627408517;=&#3627409358;
??????&#3627408518;&#3627408535;&#3627408518;&#3627408531;&#3627408532;&#3627408514;&#3627408532;, &#3627408532;&#3627408518; &#3627408517;≠&#3627409358;

Perpendicular X Ortogonal

&#3627408531; ⊥&#3627408532; <=>&#3627408534; ⊥&#3627408535; =
&#3627408502;&#3627408531;&#3627408533;&#3627408528;&#3627408520;&#3627408528;&#3627408527;&#3627408514;&#3627408522;&#3627408532;, &#3627408532;&#3627408518; &#3627408531; &#3627408518; &#3627408532; &#3627408532;ã&#3627408528; &#3627408531;&#3627408518;&#3627408535;&#3627408518;&#3627408531;&#3627408532;&#3627408514;&#3627408532;
&#3627408503;&#3627408518;&#3627408531;&#3627408529;&#3627408518;&#3627408527;&#3627408517;&#3627408522;&#3627408516;&#3627408534;&#3627408525;&#3627408514;&#3627408531;&#3627408518;&#3627408532;, &#3627408532;&#3627408518; &#3627408531; &#3627408518; &#3627408532; &#3627408532;ã&#3627408528; &#3627408516;&#3627408528;&#3627408527;&#3627408516;&#3627408528;&#3627408531;&#3627408531;&#3627408518;&#3627408527;&#3627408533;&#3627408518;&#3627408532;