Show that the points A(-2i + 3j + 5k), B(i + 2j + 3k) and C(7i - k)

Example 21 - Chapter 10 Class 12 Vector Algebra - Part 2

Example 21 - Chapter 10 Class 12 Vector Algebra - Part 3 Example 21 - Chapter 10 Class 12 Vector Algebra - Part 4

Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Transcript

Example 21 (Introduction) Show that the points A(−2𝑖 ̂ + 3𝑗 ̂ + 5𝑘 ̂), B(𝑖 ̂ + 2𝑗 ̂ + 3𝑘 ̂) and C(7𝑖 ̂ − 𝑘 ̂) are collinear. (1) Three points collinear i.e. AB + BC = AC (2) Three position vectors collinear i.e. |(𝐴𝐵) ⃗ | + |(𝐵𝐶) ⃗ | = |(𝐴𝐶) ⃗ | Example 21 Show that the points A(−2𝑖 ̂ + 3𝑗 ̂ + 5𝑘 ̂), B(𝑖 ̂ + 2𝑗 ̂ + 3𝑘 ̂) and C(7𝑖 ̂ − 𝑘 ̂) are collinear. Given A (−2𝑖 ̂ + 3𝑗 ̂ + 5𝑘 ̂) B (1𝑖 ̂ + 2𝑗 ̂ + 3𝑘 ̂) C (7𝑖 ̂ + 0𝑗 ̂ − 1𝑘 ̂) 3 points A, B, C are collinear if |(𝑨𝑩) ⃗ | + |(𝑩𝑪) ⃗ | = |(𝑨𝑪) ⃗ | Finding (𝑨𝑩) ⃗ , (𝑩𝑪) ⃗ , (𝑨𝑪) ⃗ (𝑨𝑩) ⃗ = (1 – (-2)) 𝑖 ̂ + (2 − 3) 𝑗 ̂ + (3 − 5) 𝑘 ̂ = 3𝒊 ̂ – 1𝒋 ̂ – 2𝒌 ̂ (𝑩𝑪) ⃗ = (7 − 1) 𝑖 ̂ + (0 − 2) 𝑗 ̂ + (-1−3) 𝑘 ̂ = 6𝒊 ̂ – 2𝒋 ̂ – 4𝒌 ̂ (𝑨𝑪) ⃗ = (7 − (-2)) 𝑖 ̂ + (0 − 3) 𝑗 ̂ + (-1 − 5) 𝑘 ̂ = 9𝒊 ̂ – 3𝒋 ̂ – 6𝒌 ̂ Magnitude of |(𝐴𝐵) ⃗ | = √(3^2+(−1)^2+(−2)^2 ) |(𝑨𝑩) ⃗ | = √(9+1+4) = √𝟏𝟒 Magnitude of |(𝐵𝐶) ⃗ | = √(6^2+(−2)^2+(−4)^2 ) |(𝑩𝑪) ⃗ | = √(36+4+16) = √56 = √(4 × 14) = 2√𝟏𝟒 Magnitude of |(𝐴𝐶) ⃗ | = √(9^2+(−3)^2+(−6)^2 ) |(𝑨𝑪) ⃗ | = √(81+9+36) = √126 = √(9 × 14) = 3√𝟏𝟒 Thus, |(𝑨𝑩) ⃗ | + |(𝑩𝑪) ⃗ | = √14 + 2√14 = 3√14 = |(𝑨𝑪) ⃗ | Thus, A, B and C are collinear.

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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.