The discussion revolves around a problem involving a parameterized curve described by a function \(\omega: I \rightarrow \mathbb{R}^3\) and its relationship with a constant velocity vector \(|v| = 1\). Participants are tasked with demonstrating an inequality involving the dot product of the difference of the curve's endpoints and the velocity vector, as well as an integral of the curve's derivative.
Some participants have provided hints and guidance regarding the application of calculus principles, while others are seeking clarification on the relationships between the vectors involved. The discussion is ongoing, with multiple interpretations being explored, particularly concerning the nature of the velocity vector and its derivatives.
Participants are working under the constraints of a homework assignment, which requires them to show specific inequalities and relationships without providing complete solutions. There is an emphasis on clarifying definitions and assumptions related to the vectors involved.
gabbagabbahey said:Sure, but all you really need is
\frac{\vec{t}-\vec{s}}{|\vec{t}-\vec{s}|} \cdot (\vec{t}-\vec{s}) \leq \int_a^b ||\vec{\omega}(t)'|| dt