Final Answer:
The question provides insufficient information to calculate the vectors \( \mathbf{t} \), \( \mathbf{n} \), and \( \mathbf{b} \) accurately. These vectors are typically associated with a parametric curve or a function in three-dimensional space.
Step-by-step explanation:
To determine the vectors \( \mathbf{t} \), \( \mathbf{n} \), and \( \mathbf{b} \) at a given point, the parametric representation of a curve \( r(t) \) in three-dimensional space is needed. These vectors are often calculated using the derivatives of \( r(t) \) at a specific \( t \) value.
The vectors \( \mathbf{t} \), \( \mathbf{n} \), and \( \mathbf{b} \) represent the tangent, normal, and binormal vectors, respectively, and are crucial in the study of curves in three-dimensional space. The calculations involve the derivatives of the position vector \( r(t) \) with respect to \( t \).
In this case, the lack of information regarding the specific form of \( r(t) \) hinders the ability to provide an accurate calculation. A valid response would require the explicit parametric equation or representation of the curve in question. This explanation emphasizes the importance of clarity and completeness in mathematical questions to enable precise and meaningful solutions.