a ^ . b ^ = 1 2 \widehat a\,.\,\widehat b = {1 \over {\sqrt 2 }} a . b = 2 1 and ∣ a ^ × b ^ ∣ = 1 2 |\widehat a \times \widehat b| = {1 \over {\sqrt 2 }} ∣ a × b ∣ = 2 1
( a ^ + b ^ ) . ( a ^ + 2 b ^ + 2 ( a ^ × b ^ ) ) ∣ a ^ + b ^ ∣ ∣ a ^ + 2 b ^ + 2 ( a ^ × b ^ ) ∣ = cos θ {{\left( {\widehat a + \widehat b} \right)\,.\,\left( {\widehat a + 2\widehat b + 2\left( {\widehat a \times \widehat b} \right)} \right)} \over {\left| {\widehat a + \widehat b} \right|\left| {\widehat a + 2\widehat b + 2\left( {\widehat a \times \widehat b} \right)} \right|}} = \cos \theta ∣ a + b ∣ ∣ a + 2 b + 2 ( a × b ) ∣ ( a + b ) . ( a + 2 b + 2 ( a × b ) ) = cos θ
⇒ cos θ = 1 + 3 a ^ b ^ + 2 ∣ a ^ + b ^ ∣ ∣ a ^ + 2 b ^ + 2 ( a ^ × b ^ ) ∣ \Rightarrow \cos \theta = {{1 + 3\widehat a\widehat b + 2} \over {|\widehat a + \widehat b||\widehat a + 2\widehat b + 2(\widehat a \times \widehat b)|}} ⇒ cos θ = ∣ a + b ∣∣ a + 2 b + 2 ( a × b ) ∣ 1 + 3 a b + 2
∣ a ^ + b ^ ∣ 2 = 2 + 2 |\widehat a + \widehat b{|^2} = 2 + \sqrt 2 ∣ a + b ∣ 2 = 2 + 2
∣ a ^ + 2 b ^ + 2 ( a ^ × b ^ ) ∣ 2 = 1 + 4 + 4 ∣ a ^ × b ^ ∣ 2 + 4 a ^ b ^ |\widehat a + 2\widehat b + 2(\widehat a \times \widehat b){|^2} = 1 + 4 + 4|\widehat a \times \widehat b{|^2} + 4\widehat a\widehat b ∣ a + 2 b + 2 ( a × b ) ∣ 2 = 1 + 4 + 4∣ a × b ∣ 2 + 4 a b
= 5 + 4 . 1 2 + 4 2 = 7 + 2 2 = 5 + 4\,.\,{1 \over 2} + {4 \over {\sqrt 2 }} = 7 + 2\sqrt 2 = 5 + 4 . 2 1 + 2 4 = 7 + 2 2
So, cos 2 θ = ( 3 + 3 2 ) 2 ( 2 + 2 ) ( 7 + 2 2 ) = 9 2 ( 5 2 + 3 ) 164 {\cos ^2}\theta = {{{{\left( {3 + {3 \over {\sqrt 2 }}} \right)}^2}} \over {(2 + \sqrt 2 )(7 + 2\sqrt 2 )}} = {{9\sqrt 2 (5\sqrt 2 + 3)} \over {164}} cos 2 θ = ( 2 + 2 ) ( 7 + 2 2 ) ( 3 + 2 3 ) 2 = 164 9 2 ( 5 2 + 3 )
⇒ 164 cos 2 θ = 90 + 27 2 \Rightarrow 164{\cos ^2}\theta = 90 + 27\sqrt 2 ⇒ 164 cos 2 θ = 90 + 27 2