In the following exercise, find T(t), N(t), aT_{\mathrm{T}}, and aN_{\mathrm{N}} at the given time t for the space curve r(t). [Hint: Find a(t), T(t), and aN_{\mathrm{N}}. Solve for N in the equation a(t)=aT_{\mathrm{T}}T + aN_{\mathrm{N}}N.]

Function: r(t)=etsinti+etcostj+etk\mathbf{r}(t)=e^t \sin t \mathbf{i}+e^t \cos t \mathbf{j}+e^t \mathbf{k}, Time: t=0t=0


Answered 1 year ago
Answered 1 year ago
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To find the unit tangent vector, T(t)\textbf T(t), first find the velocity function. Unit tangent vector is the quotinet of the velocity function and the magnitude of the velocity function, as shown in the equation below.

T(t)=v(t)v(t)\textbf T(t)=\frac{\textbf v(t)}{|\textbf v(t)|}

The principal unit normal vector, N(t)\textbf N(t), is the quotient of the derivative of the unit tangent vector and the magnitude of that derivative, as shown in the equation below.

N(t)=T(t)T(t)\textbf N(t)=\frac{\textbf T'(t)}{|\textbf T'(t)|}

Use the velocity function once again to find the acceleration function a(t)\textbf a(t). Derivate the obtained velocity function with respect to variable tt. Use the equation below.

a(t)=ddtv(t)\textbf a(t)=\frac{d}{dt}\textbf v(t)

Use the obtained acceleration function to find the terms below.

aT=a(t)T(t)aN=a(t)N(t)\begin{align*} a_T&=\textbf a(t)\cdot \textbf T(t)\\ a_N&=\textbf a(t)\cdot \textbf N(t)\\ \end{align*}

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