Suppose r'(t) has units of m/s. Explain why T(t) = r'(t)/ | r'(t) | is dimensionless (has no units) and carries information only about direction.

Suppose r'(t) has units of m/s.

To explain why $$\begin{gathered} \text{Let r(t) be a differential vector valued function} \\ \text{Now we look for the derivative r'(t)} \\ \text{The vector r'(t) points in the direction of travel and its length } \\ \text{tells us the speed of travel} \\ \text{Now we if we are interested in finding the direction of r' but not } \\ \text{its length } \\ \text{The standard unit tangent vector denoted by T(t) is given by:} \\ T\left(t\right)=\frac{r\text{'}\left(t\right)}{\left|r\text{'}\left(t\right)\right|} \end{gathered}$$ is dimensionless (has no units) and carries information only about direction.