Chapter 10 Class 12 Vector Algebra
Ex 10.2, 9
Ex 10.2, 10 Important
Ex 10.2, 13 Important
Ex 10.2, 17 Important
Example 14 Important
Example 16 Important
Example 21 Important
Ex 10.3, 2
Ex 10.3, 3 Important
Ex 10.3, 10 Important
Ex 10.3, 13 Important
Ex 10.3, 16 Important
Example 23 Important
Example 24
Example 25 Important
Ex 10.4, 2 Important
Ex 10.4, 5 Important
Ex 10.4, 9 Important
Ex 10.4, 10
Ex 10.4, 11 (MCQ) Important
Example 28 Important
Example 29 Important
Example 30 Important
Misc 6
Misc 12 Important
Misc 13 You are here
Misc 15 Important
Misc 19 (MCQ) Important
Chapter 10 Class 12 Vector Algebra
Last updated at April 16, 2024 by Teachoo
Misc 13 The scalar product of the vector 𝑖 ̂ + 𝑗 ̂ + 𝑘 ̂ with a unit vector along the sum of vectors 2𝑖 ̂ + 4𝑗 ̂ − 5𝑘 ̂ and λ𝑖 ̂ + 2𝑗 ̂ + 3𝑘 ̂ is equal to one. Find the value of λ. Let 𝒂 ⃗ = 𝑖 ̂ + 𝑗 ̂ + 𝑘 ̂ 𝒃 ⃗ = 2𝑖 ̂ + 4𝑗 ̂ – 5𝑘 ̂ 𝒄 ⃗ = 𝜆 𝑖 ̂ + 2𝑗 ̂ + 3𝑘 ̂ (𝒃 ⃗ + 𝒄 ⃗) = (2 + 𝜆) 𝑖 ̂ + (4 + 2) 𝑗 ̂ + (−5 + 3) 𝑘 ̂ = (2 + 𝜆) 𝒊 ̂ + 6𝒋 ̂ − 2𝒌 ̂ 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 ̂ + 6𝑗 ̂ − 2𝑘 ̂) 𝑟 ̂ = 1/√(2^2 + 𝜆^2 + 4𝜆 + 36 + 4) × ((2 + 𝜆) 𝑖 ̂ + 6𝑗 ̂ − 2𝑘 ̂) 𝒓 ̂ = 𝟏/√(𝝀^𝟐 + 𝟒𝝀 +𝟒𝟒) × ((2 + 𝜆) 𝒊 ̂ + 6𝒋 ̂ − 2𝒌 ̂) Given, 𝒂 ⃗. (𝒓 ̂) = 1 (1𝑖 ̂ + 1𝑗 ̂ + 1𝑘 ̂). (1/√(𝜆^2 + 4𝜆 +44) " × ((2 + 𝜆) " 𝑖 ̂" + 6" 𝑗 ̂" − 2" 𝑘 ̂")" ) = 1 1/√(𝜆^2 + 4𝜆 +44) (1𝑖 ̂ + 1𝑗 ̂ + 1𝑘 ̂).((𝜆 +2) 𝑖 ̂ + 6𝑗 ̂ − 2𝑘 ̂) = 1 (1𝑖 ̂ + 1𝑗 ̂ + 1𝑘 ̂).((𝜆 +2) 𝑖 ̂ + 6𝑗 ̂ − 2𝑘 ̂) = √(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 𝜆2 + 36 + 12𝜆 = 𝜆^2 + 4𝜆 +44 8𝜆 = 8 𝜆 = 8/8 𝜆 = 1 So, 𝜆 = 1