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Misc 17 - Let a and b be two unit vectors. Then a + b is a unit vector

Misc 17 - Chapter 10 Class 12 Vector Algebra - Part 2
Misc 17 - Chapter 10 Class 12 Vector Algebra - Part 3

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Misc 17 Let π‘Ž βƒ— and 𝑏 βƒ— be two unit vectors and ΞΈ is the angle between them. Then π‘Ž βƒ— + 𝑏 βƒ— is a unit vector if (A) ΞΈ = πœ‹/4 (B) ΞΈ = πœ‹/3 (C) ΞΈ = πœ‹/2 (D) ΞΈ = 2πœ‹/3 Given π‘Ž βƒ— & 𝑏 βƒ— are unit vectors, So, |𝒂 βƒ— | = 1 & |𝒃 βƒ— | = 1 We need to find ΞΈ if 𝒂 βƒ— + 𝒃 βƒ— is a unit vector Assuming π‘Ž βƒ— + 𝑏 βƒ— is a unit vector Magnitude of 𝒂 βƒ— + 𝒃 βƒ— = 1 |π‘Ž βƒ—+𝑏 βƒ— |=1 |𝒂 βƒ—+𝒃 βƒ— |^𝟐=1^2 (𝒂 βƒ—+𝒃 βƒ— ).(𝒂 βƒ—+𝒃 βƒ— ) = 1 π‘Ž βƒ—. (π‘Ž βƒ—+𝑏 βƒ— ) + 𝑏 βƒ—. (π‘Ž βƒ—+𝑏 βƒ— ) = 1 𝒂 βƒ— . 𝒂 βƒ— + π‘Ž βƒ— . 𝑏 βƒ— + 𝑏 βƒ—.π‘Ž βƒ— + 𝒃 βƒ—.𝒃 βƒ— = 1 |𝒂 βƒ— |^𝟐 + π‘Ž βƒ—.𝑏 βƒ— + 𝑏 βƒ—.π‘Ž βƒ— + |𝒃 βƒ— |^𝟐=1 12 + π‘Ž βƒ—. 𝑏 βƒ— + 𝑏 βƒ—.π‘Ž βƒ— + 12 = 1 2 + π‘Ž βƒ—. 𝑏 βƒ— + 𝒃 βƒ—.𝒂 βƒ— = 1 2 + π‘Ž βƒ—. 𝑏 βƒ— + 𝒂 βƒ—. 𝒃 βƒ— = 1 2 + 2 π‘Ž βƒ—. 𝑏 βƒ— = 1 2π‘Ž βƒ—. 𝑏 βƒ— = 1 βˆ’ 2 π‘Ž βƒ—. 𝑏 βƒ— = (βˆ’1)/2 |π‘Ž βƒ— ||𝑏 βƒ— | cos⁑ ΞΈ = (βˆ’1)/2 1 Γ— 1 Γ— cos ΞΈ = (βˆ’1)/2 cos ΞΈ = (βˆ’πŸ)/𝟐 So, ΞΈ = πŸπ…/πŸ‘ Hence, option (D) is correct Rough We know that cos 60Β° = 1/2 & cos is negative in 2nd quadrant, So, ΞΈ = 180 βˆ’ 60Β° ΞΈ = 120Β° ΞΈ = 120 Γ— πœ‹/180 ΞΈ = 2πœ‹/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 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.