Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class

Miscellaneous

Misc 1
Important

Misc 2

Misc 3 Important

Misc 4

Misc 5 Important

Misc 6

Misc 7 Important

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 You are here

Misc 18 (MCQ) Important

Misc 19 (MCQ) Important

Last updated at May 29, 2023 by Teachoo

Misc 17 Let 𝑎 ⃗ and 𝑏 ⃗ be two unit vectors and θ is the angle between them. Then 𝑎 ⃗ + 𝑏 ⃗ is a unit vector if (A) θ = 𝜋/4 (B) θ = 𝜋/3 (C) θ = 𝜋/2 (D) θ = 2𝜋/3 Given 𝑎 ⃗ & 𝑏 ⃗ are unit vectors, So, |𝒂 ⃗ | = 1 & |𝒃 ⃗ | = 1 We need to find θ if 𝒂 ⃗ + 𝒃 ⃗ is a unit vector Assuming 𝑎 ⃗ + 𝑏 ⃗ is a unit vector Magnitude of 𝒂 ⃗ + 𝒃 ⃗ = 1 |𝑎 ⃗+𝑏 ⃗ |=1 |𝒂 ⃗+𝒃 ⃗ |^𝟐=1^2 (𝒂 ⃗+𝒃 ⃗ ).(𝒂 ⃗+𝒃 ⃗ ) = 1 𝑎 ⃗. (𝑎 ⃗+𝑏 ⃗ ) + 𝑏 ⃗. (𝑎 ⃗+𝑏 ⃗ ) = 1 𝒂 ⃗ . 𝒂 ⃗ + 𝑎 ⃗ . 𝑏 ⃗ + 𝑏 ⃗.𝑎 ⃗ + 𝒃 ⃗.𝒃 ⃗ = 1 |𝒂 ⃗ |^𝟐 + 𝑎 ⃗.𝑏 ⃗ + 𝑏 ⃗.𝑎 ⃗ + |𝒃 ⃗ |^𝟐=1 12 + 𝑎 ⃗. 𝑏 ⃗ + 𝑏 ⃗.𝑎 ⃗ + 12 = 1 2 + 𝑎 ⃗. 𝑏 ⃗ + 𝒃 ⃗.𝒂 ⃗ = 1 2 + 𝑎 ⃗. 𝑏 ⃗ + 𝒂 ⃗. 𝒃 ⃗ = 1 2 + 2 𝑎 ⃗. 𝑏 ⃗ = 1 2𝑎 ⃗. 𝑏 ⃗ = 1 − 2 𝑎 ⃗. 𝑏 ⃗ = (−1)/2 |𝑎 ⃗ ||𝑏 ⃗ | cos θ = (−1)/2 1 × 1 × cos θ = (−1)/2 cos θ = (−𝟏)/𝟐 So, θ = 𝟐𝝅/𝟑 Hence, option (D) is correct Rough We know that cos 60° = 1/2 & cos is negative in 2nd quadrant, So, θ = 180 − 60° θ = 120° θ = 120 × 𝜋/180 θ = 2𝜋/3