1. **Problem statement:** Find the unit tangent vector, unit normal vector, and curvature of the circle defined by $$x = a \cos \theta, y = a \sin \theta, z = 0$$ at the point with parameter $$\theta = 0$$. 2. **Recall formulas:** - The unit tangent vector $$\mathbf{T}$$ is $$\mathbf{T} = \frac{\mathbf{r}'(\theta)}{|\mathbf{r}'(\theta)|}$$. - The unit normal vector $$\mathbf{N}$$ is $$\mathbf{N} = \frac{\mathbf{T}'(\theta)}{|\mathbf{T}'(\theta)|}$$. - The curvature $$\kappa$$ is $$\kappa = \frac{|\mathbf{T}'(\theta)|}{|\mathbf{r}'(\theta)|}$$. 3. **Find the position vector:** $$\mathbf{r}(\theta) = \langle a \cos \theta, a \sin \theta, 0 \rangle$$ 4. **Compute the first derivative:** $$\mathbf{r}'(\theta) = \langle -a \sin \theta, a \cos \theta, 0 \rangle$$ 5. **Evaluate at $$\theta=0$$:** $$\mathbf{r}'(0) = \langle 0, a, 0 \rangle$$ 6. **Compute the magnitude:** $$|\mathbf{r}'(0)| = \sqrt{0^2 + a^2 + 0^2} = a$$ 7. **Unit tangent vector:** $$\mathbf{T}(0) = \frac{\mathbf{r}'(0)}{|\mathbf{r}'(0)|} = \frac{\langle 0, a, 0 \rangle}{a} = \langle 0, 1, 0 \rangle$$ 8. **Compute the second derivative:** $$\mathbf{r}''(\theta) = \langle -a \cos \theta, -a \sin \theta, 0 \rangle$$ 9. **Evaluate at $$\theta=0$$:** $$\mathbf{r}''(0) = \langle -a, 0, 0 \rangle$$ 10. **Compute $$\mathbf{T}'(\theta)$$:** Since $$\mathbf{T} = \frac{\mathbf{r}'}{|\mathbf{r}'|}$$ and $$|\mathbf{r}'|$$ is constant $$a$$, then $$\mathbf{T}'(\theta) = \frac{\mathbf{r}''(\theta)}{a}$$ 11. **Evaluate $$\mathbf{T}'(0)$$:** $$\mathbf{T}'(0) = \frac{\langle -a, 0, 0 \rangle}{a} = \langle -1, 0, 0 \rangle$$ 12. **Magnitude of $$\mathbf{T}'(0)$$:** $$|\mathbf{T}'(0)| = \sqrt{(-1)^2 + 0^2 + 0^2} = 1$$ 13. **Unit normal vector:** $$\mathbf{N}(0) = \frac{\mathbf{T}'(0)}{|\mathbf{T}'(0)|} = \langle -1, 0, 0 \rangle$$ 14. **Curvature:** $$\kappa = \frac{|\mathbf{T}'(0)|}{|\mathbf{r}'(0)|} = \frac{1}{a}$$ **Final answers:** - Unit tangent vector at $$\theta=0$$: $$\mathbf{T} = \langle 0, 1, 0 \rangle$$ - Unit normal vector at $$\theta=0$$: $$\mathbf{N} = \langle -1, 0, 0 \rangle$$ - Curvature: $$\kappa = \frac{1}{a}$$