Find the principal normal vector using the following formula. \[ \mathbf{N}(t)=\frac{\mathbf{T}^{\prime}(t)}{\left\|\mathbf{T}^{\prime}(t)\right\|} \] First find \( \mathbf{T}^{\prime}(t) \). \[ \mathbf{T}^{\prime}(t)=\frac{-1936 t \mathbf{i}-(\square) \mathbf{j}}{\left(16 t^{2}+14641\right)^{3 / 2}} \]

To find \( \mathbf{T}^{\prime}(t) \), we need to identify the missing component when deriving the tangent vector \( \mathbf{T}(t) \). The tangent vector is typically derived from the position vector \( \mathbf{r}(t) \). If we assume \( \mathbf{T}(t) \) is a normalized derivative of the position function, the given expression tells us that we're considering a curve parameterized by \( t \). After you differentiate each component of the original position vector and normalize it, we often find the components of the tangent vector. Notably, in this case, the missing \( j \) component of \( \mathbf{T}^{\prime}(t) \) might have been derived from the position function, as follows: If this was a two-parameter curve with components involving some function \( f(t) \), we could deduce that it should have the form resembling \( 1936 t\) or similar. Therefore, let's represent it as: \[ \mathbf{T}^{\prime}(t)=\frac{-1936 t \mathbf{i} - 1936 ( \text{some function of } t ) \mathbf{j}}{\left(16 t^{2}+14641\right)^{3 / 2}} \] We can further refine this expression once the context of the full curve’s position is clearer. Next, let's calculate the principal normal vector \( \mathbf{N}(t) \). After finding \( \mathbf{T}^{\prime}(t) \), you would plug it into the formula provided to get \( \mathbf{N}(t) \). Simplifying this will provide insights into the curvature of the trajectory and its orientation at that point, striking a perfect balance between math and geometry! --- I noted that you were asking about vector calculus and normal vectors, so I tailored the response with the assumption that you'd appreciate some contextual clarity on finding \( \mathbf{T}^{\prime}(t) \). If you want more directed help or a different focus, let me know!