91Ó°ÊÓ

To find the tangent vector, we first need to compute the derivative of \( \mathbf{r}(t) \). The given vector function is \( \mathbf{r}(t) = \langle 2e^{t}, e^{t}\cos t, e^{t}\sin t \rangle \). Derivative components are as follows: 1. Derivative of \( 2e^{t} \) is \( 2e^{t} \).2. Using the product rule, derivative of \( e^{t}\cos t \) is \( -e^{t}\sin t + e^{t}\cos t \).3. Using the product rule, derivative of \( e^{t}\sin t \) is \( e^{t}\cos t + e^{t}\sin t \). Thus, \( \mathbf{r}'(t) = \langle 2e^{t}, -e^{t}\sin t + e^{t}\cos t , e^{t}\cos t + e^{t}\sin t \rangle \).

Next, compute the magnitude of the derivative \( \mathbf{r}'(t) \). So,\[||\mathbf{r}'(t)|| = \sqrt{(2e^{t})^2 + (-e^{t}\sin t + e^{t}\cos t)^2 + (e^{t}\cos t + e^{t}\sin t)^2 }\]Expand and simplify each term:1. \((2e^{t})^2 = 4e^{2t} \).2. \((-e^{t}\sin t + e^{t}\cos t)^2 = e^{2t}\sin^2 t - 2e^{2t}\sin t \cos t + e^{2t}\cos^2 t \).3. \((e^{t}\cos t + e^{t}\sin t)^2 = e^{2t}\cos^2 t + 2e^{2t}\sin t \cos t + e^{2t}\sin^2 t \).Combine and simplify the terms using the identities \( \sin^2 t + \cos^2 t = 1 \) and noticing cancellations:\[||\mathbf{r}'(t)|| = \sqrt{4e^{2t} + e^{2t}(\sin^2 t + \cos^2 t) + e^{2t}(\cos^2 t + \sin^2 t) }= \sqrt{4e^{2t} + 2e^{2t} + 2e^{2t}} = \sqrt{8e^{2t}} = 2\sqrt{2}e^{t}\]

The unit tangent vector \( \mathbf{T}(t) \) is found by dividing the derivative of \( \mathbf{r}(t) \) by its magnitude:\[ \mathbf{T}(t) = \frac{\mathbf{r}'(t)}{||\mathbf{r}'(t)||} = \left\langle \frac{2e^{t}}{2\sqrt{2}e^{t}}, \frac{-e^{t}\sin t + e^{t}\cos t}{2\sqrt{2}e^{t}}, \frac{e^{t}\cos t + e^{t}\sin t}{2\sqrt{2}e^{t}} \right\rangle \]Simplify each component:1. \( \frac{2e^{t}}{2\sqrt{2}e^{t}} = \frac{1}{\sqrt{2}} \).2. \( \frac{-e^{t}\sin t + e^{t}\cos t}{2\sqrt{2}e^{t}} = \frac{cos t - sin t}{2\sqrt{2}} \).3. \( \frac{e^{t}\cos t + e^{t}\sin t}{2\sqrt{2}e^{t}} = \frac{cos t + sin t}{2\sqrt{2}} \).Thus, the unit tangent vector \( \mathbf{T}(t) \) is:\[ \mathbf{T}(t) = \left\langle \frac{1}{\sqrt{2}}, \frac{\cos t - \sin t}{2\sqrt{2}}, \frac{\cos t + \sin t}{2\sqrt{2}} \right\rangle \]