Formulario_Gaal

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Formulário - Gaal 
 
 
𝑑 = �(𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2 𝑑 = �(𝑥2 − 𝑥1)2 + (𝑦2 − 𝑦1)2 + (𝑧2 − 𝑧1)2 
 
 
𝑥𝑀 = 𝑥𝐴+𝑥𝐵2 e 𝑦𝑀 = 𝑦𝐴+𝑦𝐵2 � 𝑥 𝑦 1𝑥𝐴 𝑦𝐴 1𝑥𝐵 𝑦𝐵 1� = 0 𝑑 = �𝑎𝑥0 + 𝑏𝑦0 + 𝑐
√𝑎2 + 𝑏2 � 
 
𝑦 − 𝑦0 = 𝑚 (𝑥 − 𝑥0) 𝑚 = ∆𝑦∆𝑥 𝑚 = −𝑎𝑏 𝑚 = 𝑡𝑔𝜃 
 
 (𝑥 − 𝑥𝑐)2 + (𝑦 − 𝑦𝑐)2 = 𝑟2 𝑥2 + 𝑦2 − 2𝑥𝑐𝑥 − 2𝑦𝑐𝑦 + 𝑥𝑐2 + 𝑦𝑐2 − 𝑟2 = 0 
 
𝑥2 = 4𝑝𝑦 𝐹 = ( 0, 𝑝) 𝑥2 = −4𝑝𝑦 𝐹 = (0,−𝑝) 
𝑦2 = 4𝑝𝑥 𝐹 = (𝑝, 0) 𝑦2 = −4𝑝𝑥 𝐹 = (−𝑝, 0) |𝑃𝐹1| + |𝑃𝐹2| = 2𝑎 �|𝑃𝐹1| − |𝑃𝐹2|� = 2𝑎 |𝐴1𝐴2| = 2𝑎 |𝐵1𝐵2| = 2𝑏 |𝐹1𝐹2| = 2𝑐 
 
𝑥2
𝑎2
+ 𝑦2
𝑏2
= 1 𝑥2
𝑏2
+ 𝑦2
𝑎2
= 1 𝑎2 = 𝑏2 + 𝑐2 𝑒 = 𝑐
𝑎
 
 
𝑥2
𝑎2
−
𝑦2
𝑏2
= 1 𝑦2
𝑎2
−
𝑥2
𝑏2
= 1 𝑐2 = 𝑎2 + 𝑏2 
 
 |𝑢�⃗ | = �𝑥2 + 𝑦2 |𝑢�⃗ | = �𝑥2 + 𝑦2 + 𝑧2 𝐴𝐵�����⃗ = 𝐵 − 𝐴 𝑢�⃗ = 𝑣|𝑣| 
 
Se 𝑢�⃗ = (𝑥1, 𝑦1, 𝑧1) 𝑒 𝑣 = (𝑥2, 𝑦2, 𝑧2), temos: 𝑢�⃗ ∙ 𝑣 = |𝑢�⃗ | |𝑣| 𝑐𝑜𝑠𝜃 |𝑢�⃗ × 𝑣| = |𝑢�⃗ | |𝑣| 𝑠𝑒𝑛𝜃 𝑢�⃗ ∙ 𝑣 = 𝑥1𝑥2 + 𝑦1𝑦2 + 𝑧1𝑧2 
𝑢�⃗ × 𝑣 = � �⃗� 𝑗 𝑘�⃗𝑥1 𝑦1 𝑧1
𝑥2 𝑦2 𝑧2� 
( 𝑢��⃗ ,𝑣��⃗ ,𝑤���⃗ ) = �𝑥1 𝑦1 𝑧1𝑥2 𝑦2 𝑧2
𝑥3 𝑦3 𝑧3
� 
𝐴 = |𝑢��⃗ × 𝑣��⃗ | 
𝑉 = |( 𝑢�⃗ , 𝑣,𝑤��⃗ )|