Check sibling questions

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


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 is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.