Math  /  Calculus

Question2 3- جد مركبات شعاع الوحدة المماسي

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The unit tangent vector UTundefined \overrightarrow{U_T} is in the direction of the velocity vector Vundefined \overrightarrow{V} .
Calculate the magnitude of Vundefined \overrightarrow{V} : Vundefined=(ω2ρ0eωt)2+(ω2ρ0eωt)2 |\overrightarrow{V}| = \sqrt{(-\omega \sqrt{2} \rho_0 e^{-\omega t})^2 + (\omega \sqrt{2} \rho_0 e^{-\omega t})^2} =2ω2ρ02e2ωt = \sqrt{2 \omega^2 \rho_0^2 e^{-2\omega t}} =ω2ρ0eωt = \omega \sqrt{2} \rho_0 e^{-\omega t}
The unit tangent vector is: UTundefined=VundefinedVundefined \overrightarrow{U_T} = \frac{\overrightarrow{V}}{|\overrightarrow{V}|} =(ω2ρ0eωt)e^ρ+(ω2ρ0eωt)e^θω2ρ0eωt = \frac{(-\omega \sqrt{2} \rho_0 e^{-\omega t}) \hat{e}_\rho + (\omega \sqrt{2} \rho_0 e^{-\omega t}) \hat{e}_\theta}{\omega \sqrt{2} \rho_0 e^{-\omega t}} =e^ρ+e^θ = -\hat{e}_\rho + \hat{e}_\theta
The components of the unit tangent vector UTundefined \overrightarrow{U_T} are:
(1,1) \boxed{(-1, 1)}

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