c → = α a → + β b → … . . ( 1 ) a → ⋅ c → = α a → ⋅ a → + β b → ⋅ a → 0 = α + β cos 1 5 ∘ … . ( 2 ) ( 1 ) ⇒ b → ⋅ c → = α a → ⋅ b → + β b → ⋅ b → ⇒ cos 7 5 ∘ = α cos 1 5 ∘ + β … . ( 3 ) ( 2 ) ( 3 ) ⇒ cos 7 5 ∘ = − β cos 2 1 5 ∘ + β \begin{aligned} & \overrightarrow{\mathrm{c}}=\alpha \overrightarrow{\mathrm{a}}+\beta \overrightarrow{\mathrm{b}} \ldots . .(1) \\ & \overrightarrow{\mathrm{a}} \cdot \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} c = α a + β b … .. ( 1 ) a ⋅ c = α a ⋅ a + β b ⋅ a 0 = α + β cos 1 5 ∘ … . ( 2 ) ( 1 ) ⇒ b ⋅ c = α a ⋅ b + β b ⋅ b ⇒ cos 7 5 ∘ = α cos 1 5 ∘ + β … . ( 3 ) ( 2 ) ( 3 ) ⇒ cos 7 5 ∘ = − β cos 2 1 5 ∘ + β \beta=\frac{\cos 75^{\circ}}{\sin ^2 15^{\circ}}=\frac{1}{\sin 15^{\circ}}=\frac{2 \sqrt{2}}{\sqrt{3}-1}
( 2 ) (2) ( 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}}
N o w Now N o w α + 2 ( 3 − 1 ) β = − ( 3 + 1 ) ( 3 − 1 ) + 2 ( 3 − 1 ) ⋅ 2 2 3 − 1 = − ( 3 + 1 ) 2 2 + 4 = − 3 − 1 − 2 3 + 8 2 = 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} α + 2 ( 3 − 1 ) β = ( 3 − 1 ) − ( 3 + 1 ) + 3 − 1 2 ( 3 − 1 ) ⋅ 2 2 = 2 − ( 3 + 1 ) 2 + 4 = 2 − 3 − 1 − 2 3 + 8 = 2 − 3 $
