Question 10 (OR 2 nd question)

Find the angle between the vectors:

  a = i + j – k and b = i – j + k

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Question 10 (OR 2nd question) Find the angle between the vectors: π‘Ž βƒ— = 𝑖 Μ‚ + 𝑗 Μ‚ – π‘˜ Μ‚ and 𝑏 βƒ— = 𝑖 Μ‚ – 𝑗 Μ‚ + π‘˜ Μ‚ Given π‘Ž βƒ— = 𝑖 Μ‚ + 𝑗 Μ‚ – π‘˜ Μ‚ and 𝑏 βƒ— = 𝑖 Μ‚ – 𝑗 Μ‚ + π‘˜ Μ‚ We know that π‘Ž βƒ— . 𝑏 βƒ— = |π‘Ž βƒ— ||𝑏 βƒ— | cos ΞΈ where ΞΈ is the angle between π‘Ž βƒ— & 𝑏 βƒ— Now, 𝒂 βƒ—. 𝒃 βƒ— = (𝑖 Μ‚ + 𝑗 Μ‚ – π‘˜ Μ‚). (𝑖 Μ‚ – 𝑗 Μ‚ + π‘˜ Μ‚) = (1𝑖 Μ‚ + 1𝑗 Μ‚ – π‘˜ Μ‚). (1𝑖 Μ‚ βˆ’ 1𝑗 Μ‚ + 1π‘˜ Μ‚) = 1 Γ— 1 + 1 Γ— (βˆ’1) + (-1) Γ— 1 = 1 – 1 – 1 = –1 Magnitude of π‘Ž βƒ— = √(12+12+(βˆ’1)2) |π‘Ž βƒ— |= √(1+1+1) = √3 Magnitude of 𝑏 βƒ— = √(12+(βˆ’1)2+12) |𝑏 βƒ— |= √(1+1+1) = √3 Now, π‘Ž βƒ— . 𝑏 βƒ— = |π‘Ž βƒ— ||𝑏 βƒ— | cos ΞΈ –1 = √3 Γ— √3 Γ— cos ΞΈ –1 = 3 cos ΞΈ cos ΞΈ = (βˆ’1)/3 ΞΈ = 𝒄𝒐𝒔^(βˆ’πŸ) ((βˆ’πŸ)/πŸ‘) Thus, the angle between π‘Ž βƒ— and 𝑏 βƒ— is cos-1((βˆ’πŸ)/πŸ‘)

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.