Omar E.

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Mr Mohamed Shehata Geometry Senior 1 Geome try ➢ OPERATION ON VETORS ➢ VECTORS • Triangle Rule: AB + BC = AC 𝑨ሬሬሬሬ𝑩ሬሬԦ = B – A B ||AB|| = ||BA|| 𝑨ሬሬሬሬ𝑩ሬሬԦ ≠ 𝑩ሬሬሬሬሬ𝑨ሬԦ 𝑨ሬሬሬሬ𝑩ሬሬԦ , 𝑩ሬሬሬሬሬ𝑨ሬԦ equal in magnitude but opposite in direction A C 𝒌൫𝑨ሬሬറ+ 𝑩ሬሬറ൯ = 𝒌𝑨ሬሬറ+ 𝑲𝑩ሬሬറ • AB + AC = 2AD (AD Median) ||k𝑨ሬሬറ||=ȁ𝒌ȁ.||𝑨ሬሬറ|| A 𝑨ሬሬሬሬ𝑩ሬሬԦ = (x , y) , ||AB|| = ඥ𝒙𝟐 + 𝒚𝟐 𝒚 Direction tan𝜽 = 𝒙 • FUNDAMENTAL FORM: B C D 𝒊Ƹ = (1, 0) , 𝒋Ƹ = (0 , 1) • Parallelogram Rule: 𝑨ሬሬሬሬ𝑩ሬሬറ = 𝒂𝒊Ƹ + 𝒃𝒋Ƹ • AB + AD = AC B C • CARTISAN: 𝑨ሬሬሬሬ𝑩ሬሬԦ = (x , y) • POLAR FORM: 𝑨ሬሬሬሬ𝑩ሬሬԦ = (ȁȁ𝐀𝐁ȁȁ,𝜽) • IF 𝑨ሬሬറ//𝑩ሬሬറ D A 𝒕𝒂𝒏 𝜽 = 𝒕𝒂𝒏 𝜽 𝒚 𝟐 = 𝒚 𝟏 𝟏 𝟐 Aሬሬറ Bሬሬറ Re mark 𝒙 𝟐 𝒙 𝟏 𝜽 𝟏 𝜽 𝟐 𝑨𝑩= 𝑩−𝑨 𝑨𝑩 = −𝑩𝑨 • IF 𝑨ሬሬറ ⊥ 𝑩ሬሬറ 𝒕𝒂𝒏𝜽 × 𝒕𝒂𝒏𝜽 = −𝟏 ➢ DIVISION OF LINE SEGMENT 𝟏 𝟐 Aሬሬറ Bሬሬറ 𝒚 𝒚 𝟏 × 𝟐 = −𝟏 𝜽 𝒙 +𝒙 𝒚 +𝒚 𝜽 𝟐 • Midpoint: 𝒄(𝒙,𝒚) = ቀ 𝟏 𝟐, 𝟏 𝟐ቁ 𝒙 𝒙 𝟏 𝟏 𝟐 𝟐 𝟐 Remark 𝐴(𝑥 1 ,𝑦 1 ) 𝐶(𝑥,𝑦) 𝐵(𝑥 2 ,𝑦 2 ) If 𝑨ሬሬറ = (𝒙,𝒚) ,𝒕𝒉𝒆𝒏 𝒕𝒉𝒆 𝒔𝒍𝒐 𝒑𝒆 𝒐𝒇 𝑨ሬሬറ= 𝒚 • Centroid: 𝒙 𝒙 +𝒙 +𝒙 𝒚 +𝒚 +𝒚 𝒎(𝒙,𝒚) = ቀ 𝟏 𝟐 𝟑, 𝟏 𝟐 𝟑ቁ 𝟑 𝟑 B Conversions:  From cartesian to polar 𝒃 (𝒂,𝒃) → ൭ඥ𝒂𝟐 +𝒃𝟐,𝒕𝒂𝒏−𝟏൬ ൰൱ 𝒂 𝑚(𝑥,𝑦)  From polar to cartesian (𝒙,𝜽) → (𝒙𝒄𝒐𝒔𝜽,𝒙𝒔𝒊𝒏𝜽) C A A P age 1 | 3 Mr Mohamed Shehata Geometry Senior 1 • Internally: ✓ 𝒂𝒙+ 𝒃𝒚 = 𝒄 𝒙 𝒎 +𝒙 𝒎 𝒚 𝒎 +𝒚 𝒎 𝒄𝒐𝒇𝒇𝒙 −𝒂 𝒄(𝒙,𝒚) = ቀ 𝟏 𝟏 𝟐 𝟐, 𝟏 𝟐 𝟐 𝟐ቁ Slope=− = 𝒎 +𝒎 𝒎 +𝒎 𝒄𝒐𝒇𝒇𝒚 𝒃 𝟏 𝟐 𝟏 𝟐 ✓ 𝑼ሬሬറ (𝒙,𝒚) 𝒔𝒍𝒐𝒑𝒆 = 𝒚 𝑚 2 𝑚 1 𝒙 ✓ If given : Two point on the straight line 𝒚 − 𝒚 𝐴(𝑥 1 ,𝑦 1 ) 𝐶(𝑥,𝑦) 𝐵(𝑥 2 ,𝑦 2 ) 𝒎 = 𝟐 𝟏 𝒙 − 𝒙 𝟐 𝟏 • Externally: 𝒙 𝒎 −𝒙 𝒎 𝒚 𝒎 −𝒚 𝒎 𝒄(𝒙,𝒚) = ቀ 𝟏 𝟏 𝟐 𝟐, 𝟏 𝟏 𝟐 𝟐ቁ • If L , L are two straight lines: 1 2 𝒎 −𝒎