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

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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

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Transcript

Example 21 (Introduction) Show that the points A(βˆ’2𝑖 Μ‚ + 3𝑗 Μ‚ + 5π‘˜ Μ‚), B(𝑖 Μ‚ + 2𝑗 Μ‚ + 3π‘˜ Μ‚) and C(7𝑖 Μ‚ βˆ’ π‘˜ Μ‚) are collinear. (1) Three points collinear i.e. AB + BC = AC (2) Three position vectors collinear i.e. |(𝐴𝐡) βƒ— | + |(𝐡𝐢) βƒ— | = |(𝐴𝐢) βƒ— | Example 21 Show that the points A(βˆ’2𝑖 Μ‚ + 3𝑗 Μ‚ + 5π‘˜ Μ‚), B(𝑖 Μ‚ + 2𝑗 Μ‚ + 3π‘˜ Μ‚) and C(7𝑖 Μ‚ βˆ’ π‘˜ Μ‚) are collinear. Given A (βˆ’2𝑖 Μ‚ + 3𝑗 Μ‚ + 5π‘˜ Μ‚) B (1𝑖 Μ‚ + 2𝑗 Μ‚ + 3π‘˜ Μ‚) C (7𝑖 Μ‚ + 0𝑗 Μ‚ βˆ’ 1π‘˜ Μ‚) 3 points A, B, C are collinear if |(𝑨𝑩) βƒ— | + |(𝑩π‘ͺ) βƒ— | = |(𝑨π‘ͺ) βƒ— | Finding (𝑨𝑩) βƒ— , (𝑩π‘ͺ) βƒ— , (𝑨π‘ͺ) βƒ— (𝑨𝑩) βƒ— = (1 – (-2)) 𝑖 Μ‚ + (2 βˆ’ 3) 𝑗 Μ‚ + (3 βˆ’ 5) π‘˜ Μ‚ = 3π’Š Μ‚ – 1𝒋 Μ‚ – 2π’Œ Μ‚ (𝑩π‘ͺ) βƒ— = (7 βˆ’ 1) 𝑖 Μ‚ + (0 βˆ’ 2) 𝑗 Μ‚ + (-1βˆ’3) π‘˜ Μ‚ = 6π’Š Μ‚ – 2𝒋 Μ‚ – 4π’Œ Μ‚ (𝑨π‘ͺ) βƒ— = (7 βˆ’ (-2)) 𝑖 Μ‚ + (0 βˆ’ 3) 𝑗 Μ‚ + (-1 βˆ’ 5) π‘˜ Μ‚ = 9π’Š Μ‚ – 3𝒋 Μ‚ – 6π’Œ Μ‚ Magnitude of |(𝐴𝐡) βƒ— | = √(3^2+(βˆ’1)^2+(βˆ’2)^2 ) |(𝑨𝑩) βƒ— | = √(9+1+4) = βˆšπŸπŸ’ Magnitude of |(𝐡𝐢) βƒ— | = √(6^2+(βˆ’2)^2+(βˆ’4)^2 ) |(𝑩π‘ͺ) βƒ— | = √(36+4+16) = √56 = √(4 Γ— 14) = 2βˆšπŸπŸ’ Magnitude of |(𝐴𝐢) βƒ— | = √(9^2+(βˆ’3)^2+(βˆ’6)^2 ) |(𝑨π‘ͺ) βƒ— | = √(81+9+36) = √126 = √(9 Γ— 14) = 3βˆšπŸπŸ’ Thus, |(𝑨𝑩) βƒ— | + |(𝑩π‘ͺ) βƒ— | = √14 + 2√14 = 3√14 = |(𝑨π‘ͺ) βƒ— | Thus, A, B and C are collinear.

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.