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1 point) use the inner product ⟨f,g⟩=∫10f(x)g(x)dx ⟨f,g⟩=∫01f(x)g(x)dx in the vector space c0[0,1]c0[0,1] of continuous functions on the domain [0,1][0,1] to find ⟨f,g⟩⟨f,g⟩, ‖f‖‖f‖, ‖g‖‖g‖, and the angle αf,gαf,g between f(x)f(x) and g(x)g(x) for f(x)=10x2−3 and g(x)=6x−9.

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LammettHash
[tex]f(x)=10x^2-3[/tex]
[tex]g(x)=6x-9[/tex]

[tex]\langle f,g\rangle=\displaystyle\int_0^1(10x^2-3)(6x-9)\,\mathrm dx=3[/tex]

[tex]\|f\|=\sqrt{\langle f,f\rangle}=\sqrt9=3[/tex]

[tex]\|g\|=\sqrt{\langle g,g\rangle}=\sqrt{39}[/tex]

[tex]\cos\alpha=\dfrac{\langle f,g\rangle}{\|f\|\|g\|}=\dfrac3{9\sqrt39}=\dfrac1{3\sqrt{39}}\implies \alpha=\cos^{-1}\dfrac1{3\sqrt{39}}[/tex]

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