Misc 8 - Show A, B, C are collinear, find ratio where B - Miscellaneou

Misc 8 - Chapter 10 Class 12 Vector Algebra - Part 2

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Misc 8 (Introduction) Show that the points A(1, – 2, – 8), B (5, 0, – 2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC. (1) Three points collinear i.e. AB + BC = AC (2) Three position vectors collinear i.e. |(𝐴𝐵) ⃗ | + |(𝐵𝐶) ⃗ | = |(𝐴𝐶) ⃗ | Misc 8 Show that the points A(1, – 2, – 8), B (5, 0, – 2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC. vGiven A (1, -2, −8) B (5, 0,−2) C (11, 3, 7) 3 points A, B, C are collinear if |(𝑨𝑩) ⃗ | + |(𝑩𝑪) ⃗ | = |(𝑨𝑪) ⃗ | Finding (𝑨𝑩) ⃗ , (𝑩𝑪) ⃗ , (𝑨𝑪) ⃗ (𝑨𝑩) ⃗ = (5 − 1) 𝑖 ̂ + (0 − (−2)) 𝑗 ̂ + (−2−(−8)) 𝑘 ̂ = 4𝑖 ̂ + 2𝑗 ̂ + 6𝑘 ̂ (𝑩𝑪) ⃗ = (11 − 5) 𝑖 ̂ + (3 − 0) 𝑗 ̂ + (7−(−2)) 𝑘 ̂ = 6𝒊 ̂ + 3𝒋 ̂ + 9𝒌 ̂ (𝑨𝑪) ⃗ = (11 − 1) 𝑖 ̂ + (3 − (−2)) 𝑗 ̂ + (7−(−8)) 𝑘 ̂ = 10𝒊 ̂ + 5𝒋 ̂ + 15𝒌 ̂ Magnitude of (𝐴𝐵) ⃗ = √(42+22+62) |(𝑨𝑩) ⃗ | = √(16+4+36) = √56 = √(4×14 ) = 2√𝟏𝟒 Magnitude of (𝐵𝐶) ⃗ = √(62+32+92) |(𝑩𝑪) ⃗ |= √(36+9+81) = √126 = √(9×14 ) = 3√𝟏𝟒 Magnitude of (𝐴𝐶) ⃗ = √(102+52+152) |(𝑨𝑪) ⃗ |= √(100+25+225)= √350 = √(25 × 14 ) = 5√𝟏𝟒 Thus, |(𝑨𝑩) ⃗ | + |(𝑩𝑪) ⃗ | = 2√(14 ) + 3√(14 ) = 5√(14 ) = |(𝑨𝑪) ⃗ | Thus, A, B and C are collinear. Finding the ratio in which B divides AC Let B divide AC in the ratio k : 1 Here, (𝑶𝑨) ⃗ = 1𝑖 ̂ − 2𝑗 ̂ − 8𝑘 ̂ (𝑶𝑩) ⃗ = 5𝑖 ̂ + 0𝑗 ̂ − 2𝑘 ̂ and (𝑶𝑪) ⃗ = 11𝑖 ̂ + 3𝑗 ̂ + 7𝑘 ̂ Position vector of 𝑩 = (𝒌(𝑶𝑪) ⃗ + 𝟏.(𝑶𝑨) ⃗)/(𝒌 + 𝟏) 5𝑖 ̂ + 0𝑗 ̂ − 2𝑘 ̂ = (𝑘(11𝑖 ̂ + 3𝑗 ̂ + 7𝑘 ̂ ) + 1(1𝑖 ̂ − 2𝑗 ̂ − 8𝑘 ̂ ))/(𝑘 + 1) 5𝑖 ̂ + 0𝑗 ̂ − 2𝑘 ̂ = (11𝑘𝑖 ̂ + 3𝑘𝑗 ̂ + 7𝑘 𝑘 ̂ + 𝑖 ̂ − 2𝑗 ̂ − 8𝑘 ̂)/(𝑘 + 1) 5𝒊 ̂ + 0𝒋 ̂ − 2𝒌 ̂ = ((𝟏𝟏𝒌 + 𝟏) 𝒊 ̂ + (𝟑𝒌 − 𝟐) 𝒋 ̂ + (𝟕𝒌 − 𝟖) 𝒌 ̂)/(𝒌 + 𝟏) Since the two vectors are equal, corresponding components are also equal. So, (𝟏𝟏𝒌 + 𝟏)/(𝒌 + 𝟏) = 5 11k + 1 = 5k + 5 11k – 5k = 5 − 1 6k = 4 k = 4/6 = 2/3 Thus, B divides AC in the ratio 2 : 3

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.