Miscellaneous

Misc 1
Important

Misc 2

Misc 3 Important

Misc 4

Misc 5 Important

Misc 6

Misc 7 Important You are here

Misc 8 Important

Misc 9

Misc 10

Misc 11 Important

Misc 12 Important

Misc 13

Misc 14 Important

Misc 15 Important

Misc 16 (MCQ) Important

Misc 17 (MCQ) Important

Misc 18 (MCQ) Important

Misc 19 (MCQ) Important

Last updated at May 6, 2021 by Teachoo

Misc 7 If 𝑎 ⃗ = 𝑖 ̂ + 𝑗 ̂ + 𝑘 ̂, 𝑏 ⃗ = 2𝑖 ̂ −𝑗 ̂ + 3𝑘 ̂ and 𝑐 ⃗ = 𝑖 ̂ − 2𝑗 ̂ + 𝑘 ̂ , find a unit vector parallel to the vector 2𝑎 ⃗ – 𝑏 ⃗ + 3𝑐 ⃗ . Given 𝑎 ⃗ = 𝑖 ̂ + 𝑗 ̂ + 𝑘 ̂ 𝑏 ⃗ = 2𝑖 ̂ + 𝑗 ̂ + 3𝑘 ̂ 𝑐 ⃗ = 𝑖 ̂ − 2𝑗 ̂ + 𝑘 ̂ Let 𝒓 ⃗ = 2𝒂 ⃗ − 𝒃 ⃗ + 3𝒄 ⃗ = 2(𝑖 ̂ + 𝑗 ̂ + 𝑘 ̂) − (2𝑖 ̂ − 𝑗 ̂ + 3𝑘 ̂) + 3(𝑖 ̂ −2𝑗 ̂ + 𝑘 ̂) = 2𝑖 ̂ + 2𝑗 ̂ + 2𝑘 ̂ − 2𝑖 ̂ + 1𝑗 ̂ − 3𝑘 ̂ + 3𝑖 ̂ − 6𝑗 ̂ + 3𝑘 ̂ = (2 − 2 + 3) 𝑖 ̂ + (2 + 1 − 6) 𝑗 ̂ + (2 − 3 + 3) 𝑘 ̂ = 3𝒊 ̂ – 3𝒋 ̂ + 2𝒌 ̂ ∴ 𝒓 ⃗ = 3𝒊 ̂ – 3𝒋 ̂ + 2𝒌 ̂ Magnitude of 𝑟 ⃗ = √(32+(−3)2+22) |𝒓 ⃗ | = √(9+9+4) = √𝟐𝟐 Unit vector in the direction of 𝑟 ⃗ = 𝟏/|𝒓 ⃗ | x 𝒓 ⃗ = 1/√22 × [3𝑖 ̂ −3𝑗 ̂+2𝑘 ̂ ] = 3/√22 𝑖 ̂ – 3/√22 𝑗 ̂ + 2/√22 𝑘 ̂ Hence the required vector is 𝟑/√𝟐𝟐 𝒊 ̂ – 𝟑/√𝟐𝟐 𝒋 ̂ + 𝟐/√𝟐𝟐 𝒌 ̂