1. **State the problem:** We need to find the unit tangent vector $\mathbf{T}$, the principal unit normal vector $\mathbf{N}$, and the unit binormal vector $\mathbf{B}$ for the curve $\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle$. However, the given curve is incomplete. Assuming a general curve $\mathbf{r}(t) = \langle x(t), y(t), 3k \rangle$ where $k$ is a constant. 2. **Formulas and important rules:** - The unit tangent vector is $\mathbf{T}(t) = \frac{\mathbf{r}'(t)}{\|\mathbf{r}'(t)\|}$. - The principal unit normal vector is $\mathbf{N}(t) = \frac{\mathbf{T}'(t)}{\|\mathbf{T}'(t)\|}$. - The unit binormal vector is $\mathbf{B}(t) = \mathbf{T}(t) \times \mathbf{N}(t)$. 3. **Intermediate work:** - Compute $\mathbf{r}'(t) = \langle x'(t), y'(t), 0 \rangle$ since $3k$ is constant. - Compute $\|\mathbf{r}'(t)\| = \sqrt{(x'(t))^2 + (y'(t))^2}$. - Then $\mathbf{T}(t) = \frac{\langle x'(t), y'(t), 0 \rangle}{\sqrt{(x'(t))^2 + (y'(t))^2}}$. - Next, find $\mathbf{T}'(t)$ by differentiating each component of $\mathbf{T}(t)$. - Compute $\|\mathbf{T}'(t)\|$ and then $\mathbf{N}(t) = \frac{\mathbf{T}'(t)}{\|\mathbf{T}'(t)\|}$. - Finally, compute $\mathbf{B}(t) = \mathbf{T}(t) \times \mathbf{N}(t)$ using the cross product formula. 4. **Explanation:** - The unit tangent vector $\mathbf{T}$ points in the direction of the curve's motion. - The principal unit normal vector $\mathbf{N}$ points towards the center of curvature, showing how the curve bends. - The binormal vector $\mathbf{B}$ is perpendicular to both $\mathbf{T}$ and $\mathbf{N}$, completing the right-handed orthonormal frame. Since the exact functions $x(t)$ and $y(t)$ are not provided, the vectors are expressed in terms of derivatives of $x$ and $y$.