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Section 7 Wrap Up Questions

  1. Decide whether each of the following statements is true or false. Give a reason for your answer.

    1. Any parameterization of a circle has to include trigonometric functions.
    2. The vector equations \(\vec{r} (t)=t\vec{i}+t^2\vec{j}\) and \(\vec{r}(t) =\ln{(t)} \vec{i} +(\ln{(t)})^2 \vec{j}\) parameterize the same curve.
    3. To switch the orientation of a parameterization, we can replace \(t\) by \(-t\text{.}\)
  2. Recall that a line \(L\) line passing through \(P\) in the direction of the vector \(\vec{v}\) is parameterized by \(\vec{r} (t) =\vec{P} +\vec{v} t.\) Explain how to modify this to parameterize a line between points \(Q\) and \(R\) — that is, what point and what direction vector would you choose?
  3. How would you decide whether or not a curve \(C\) parameterized by \(\vec{r}(t)=x(t)\vec{i} +y(t)\vec{j}\) passes through a given point \(P=(a,b)\text{?}\)
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