a m p ; c → = α a → + β b → … . ( 1 ) a m p ; a → . c → = α a → ⋅ a → + β b → ⋅ a → a m p ; 0 = α + β cos 15 ∘ … . ( 2 ) a m p ; ( 1 ) ⇒ b → ⋅ c → = α a → ⋅ b → + β b → ⋅ b → a m p ; ⇒ cos 75 ∘ = α cos 15 ∘ + β … . ( 3 ) a m p ; ( 2 ) & a m p ; ( 3 ) ⇒ cos 75 ∘ = − β cos 2 15 ∘ + β \begin{aligned}& \overrightarrow{\mathrm{c}}=\alpha \overrightarrow{\mathrm{a}}+\beta \overrightarrow{\mathrm{b}} \ldots .(1) \\& \overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{c}}=\alpha \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{a}}+\beta \overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{a}} \\& 0=\alpha+\beta \cos 15^{\circ} \ldots .(2) \\& (1) \Rightarrow \overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}=\alpha \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{~b}}+\beta \overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{~b}} \\& \Rightarrow \cos 75^{\circ}=\alpha \cos 15^{\circ}+\beta \ldots .(3) \\& (2) \&(3) \Rightarrow \cos 75^{\circ}=-\beta \cos ^2 15^{\circ}+\beta\end{aligned} am p ; c = α a + β b … . ( 1 ) am p ; a . c = α a ⋅ a + β b ⋅ a am p ; 0 = α + β cos 1 5 ∘ … . ( 2 ) am p ; ( 1 ) ⇒ b ⋅ c = α a ⋅ b + β b ⋅ b am p ; ⇒ cos 7 5 ∘ = α cos 1 5 ∘ + β … . ( 3 ) am p ; ( 2 ) & am p ; ( 3 ) ⇒ cos 7 5 ∘ = − β cos 2 1 5 ∘ + β
a m p ; β = cos 75 ∘ sin 2 15 ∘ = 1 sin 15 ∘ = 2 2 3 − 1 a m p ; ( 2 ) ⇒ α = − cos 15 ∘ sin 15 ∘ = − ( 3 + 1 ) ( 3 − 1 ) a m p ; ∴ c → = − ( 3 + 1 ) ( 3 − 1 ) a → + ( 2 2 3 − 1 ) b → \begin{aligned}& \beta=\frac{\cos 75^{\circ}}{\sin ^2 15^{\circ}}=\frac{1}{\sin 15^{\circ}}=\frac{2 \sqrt{2}}{\sqrt{3}-1} \\& (2) \Rightarrow \alpha=\frac{-\cos 15^{\circ}}{\sin 15^{\circ}}=\frac{-(\sqrt{3}+1)}{(\sqrt{3}-1)} \\& \therefore \overrightarrow{\mathrm{c}}=\frac{-(\sqrt{3}+1)}{(\sqrt{3}-1)} \overrightarrow{\mathrm{a}}+\left(\frac{2 \sqrt{2}}{\sqrt{3}-1}\right) \overrightarrow{\mathrm{b}}\end{aligned} am p ; β = sin 2 1 5 ∘ cos 7 5 ∘ = sin 1 5 ∘ 1 = 3 − 1 2 2 am p ; ( 2 ) ⇒ α = sin 1 5 ∘ − cos 1 5 ∘ = ( 3 − 1 ) − ( 3 + 1 ) am p ; ∴ c = ( 3 − 1 ) − ( 3 + 1 ) a + ( 3 − 1 2 2 ) b
Now
a m p ; α + 2 ( 3 − 1 ) β = − ( 3 + 1 ) ( 3 − 1 ) + 2 ( 3 − 1 ) ⋅ 2 2 3 − 1 a m p ; = − ( 3 + 1 ) 2 2 + 4 a m p ; = − 3 − 1 − 2 3 + 8 2 a m p ; = 2 − 3 \begin{aligned}& \alpha+\sqrt{2}(\sqrt{3}-1) \beta=\frac{-(\sqrt{3}+1)}{(\sqrt{3}-1)}+\frac{\sqrt{2}(\sqrt{3}-1) \cdot 2 \sqrt{2}}{\sqrt{3}-1} \\& =\frac{-(\sqrt{3}+1)^2}{2}+4 \\& =\frac{-3-1-2 \sqrt{3}+8}{2} \\& =2-\sqrt{3}\end{aligned} am p ; α + 2 ( 3 − 1 ) β = ( 3 − 1 ) − ( 3 + 1 ) + 3 − 1 2 ( 3 − 1 ) ⋅ 2 2 am p ; = 2 − ( 3 + 1 ) 2 + 4 am p ; = 2 − 3 − 1 − 2 3 + 8 am p ; = 2 − 3
