Answer
Result False
Work Step by Step
Step 1 To answer this question, we shall first try to understand what a Unit Tangent Vector is and then argue whether the statement is true or false. Step 2 In this case, since we know that the Unit Tangent Vector always points in the direction of increasing parameter, we can deem the latter part of the statement to be correct. To analyze the former part, let us first write the equation for $\mathbf{T}(t)$: \[ \mathbf{T}(t) = \frac{\mathbf{r}'(t)}{\|\mathbf{r}'(t)\|} \] The statement that needs to be examined is whether or not $\mathbf{T}(t)$ is orthogonal to $\mathbf{r}(t)$. Since $\mathbf{T}(t)$ is defined to be the unit vector of the slope of $\mathbf{r}(t)$, one can say that it is tangent to $\mathbf{r}(t)$, not orthogonal. Therefore, one can conclude that the statement in question is false. Result False
