Misc 17

Transcript

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 Therefore, 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 2 𝑎﷯. 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 (B) is correct