You’re programming a robotic arm for a manufacturing facility. The arm’s position at time [latex]t[/latex] (in seconds) is described by the vector-valued function [latex]\mathbf{r}(t)[/latex], where distances are measured in centimeters. To ensure smooth and precise movements, you need to analyze the arm’s velocity (the derivative of position), find tangent vectors that show the direction of motion at specific points, calculate unit tangent vectors for directional control, and determine the arm’s total displacement using integrals.