Let a and b be two unit vectors such that a + b + 2 a × b = 2 . If θ ∈ 0 , π is the angle between a ^ and b ^ , then among the statements: S 1 : 2 a ^ × b ^ = a ^ - b ^ S 2 : The projection of a ^ on a ^ + b ^ is 1 2

Step-by-step solution

Given, a ^ + b ^ + 2 a ^ × b ^ = 2 , θ ∈ 0 , π Squaring both side we get, a ^ + b ^ + 2 a ^ × b ^ · a ^ + b ^ + 2 a ^ × b ^ = 4 a ^ + b ^ 2 + 4 a ^ × b ^ 2 + 0 = 4 Let the angle be θ between a ^ and b ^ ⇒ a ^ 2 + b ^ 2 + 2 a ^ b ^ cos θ + 4 a ^ b ^ sin θ = 4 ⇒ 2 + 2 cos θ + 4 sin 2 θ = 4 ⇒ 2 + 2 cos θ - 4 cos 2 θ = 0 Let cos θ = t then ⇒ 2 t 2 - t - 1 = 0 ⇒ 2 t 2 - 2 t + t - 1 = 0 ⇒ 2 t t - 1 + t - 1