Misc 13 - Scalar product of vector i + j + k with unit vector - Miscellaneous

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Slide37.JPG

  1. Chapter 10 Class 12 Vector Algebra
  2. Serial order wise

Transcript

Misc 13 The scalar product of the vector 𝑖﷯ + 𝑗﷯ + 𝑘﷯ with a unit vector along the sum of vectors 2 𝑖﷯ + 4 𝑗﷯ − 5 𝑘﷯ and λ 𝑖﷯ + 2 𝑗﷯ + 3 𝑘﷯ is equal to one. Find the value of λ. Let 𝑎﷯ = 𝑖﷯ + 𝑗﷯ + 𝑘﷯ = 1 𝑖﷯ + 1 𝑗﷯ + 1 𝑘﷯ 𝑏﷯ = 2 𝑖﷯ + 4 𝑗﷯ – 5 𝑘﷯ 𝑐﷯ = 𝜆 𝑖﷯ + 2 𝑗﷯ + 3 𝑘﷯ ( 𝑏﷯ + 𝑐﷯) = (2 + 𝜆) 𝑖﷯ + (4 + 2) 𝑗﷯ + (−5 + 3) 𝑘﷯ = (2 + 𝜆) 𝑖﷯ + 6 𝑗﷯ − 2 𝑘﷯ Let 𝑟﷯ be unit vector along ( 𝑏﷯ + 𝑐﷯) 𝑟﷯ = 1﷮𝑀𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 ( 𝑏﷯ + 𝑐﷯)﷯ × ( 𝑏﷯ + 𝑐﷯) 𝑟﷯ = 1﷮ ﷮ 2 + 𝜆﷯﷮2﷯ + 6﷮2﷯ + −2﷯﷮2﷯﷯﷯ × ((2 + 𝜆) 𝑖﷯ + 6 𝑗﷯ − 2 𝑘﷯) 𝑟﷯ = 1﷮ ﷮ 2﷮2﷯ + 𝜆﷮2﷯ + 4𝜆 + 36 + 4﷯﷯ × ((2 + 𝜆) 𝑖﷯ + 6 𝑗﷯ − 2 𝑘﷯) 𝒓﷯ = 𝟏﷮ ﷮ 𝝀﷮𝟐﷯ + 𝟒𝝀 +𝟒𝟒﷯﷯ × ((2 + 𝜆) 𝒊﷯ + 6 𝒋﷯ − 2 𝒌﷯) Given, 𝑎﷯. ( 𝑟﷯) = 1 (1 𝑖﷯ + 1 𝑗﷯ + 1 𝑘﷯).( 1﷮ ﷮ 𝜆﷮2﷯ + 4𝜆 +44﷯﷯ × ((2 + 𝜆) 𝑖﷯ + 6 𝑗﷯ − 2 𝑘﷯)) = 1 1﷮ ﷮ 𝜆﷮2﷯ + 4𝜆 +44﷯﷯ (1 𝑖﷯ + 1 𝑗﷯ + 1 𝑘﷯).((𝜆 +2) 𝑖﷯ + 6 𝑗﷯ − 2 𝑘﷯) = 1 (1 𝑖﷯ + 1 𝑗﷯ + 1 𝑘﷯).((𝜆 +2) 𝑖﷯ + 6 𝑗﷯ − 2 𝑘﷯) = ﷮ 𝜆﷮2﷯ + 4𝜆 +44﷯ 1.(𝜆 + 2) + 1.6 + 1.(−2) = ﷮ 𝜆﷮2﷯ + 4𝜆 +44﷯ 𝜆 + 2 + 6 − 2 = ﷮ 𝜆﷮2﷯ + 4𝜆 +44﷯ 𝜆 + 6 = ﷮ 𝝀﷮𝟐﷯ + 𝟒𝝀 +𝟒𝟒﷯ Squaring both sides (𝜆 + 6)2 = ﷮ 𝜆﷮2﷯ + 4𝜆 +44﷯﷯﷮2﷯ (𝜆 + 6)2 = ﷮ 𝜆﷮2﷯ + 4𝜆 +44﷯﷯﷮2﷯ 𝜆2 + 36 + 12𝜆 = 𝜆﷮2﷯ + 4𝜆 +44 𝜆2 – 𝜆﷮2﷯ + 12𝜆 – 4𝜆 = 44 – 36 8𝜆 = 8 𝜆 = 8﷮8﷯ 𝜆 = 1 So, 𝜆 = 1

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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.