# Thread: Parametrised by arc length

1. ## Parametrised curves

Suppose that $\displaystyle \alpha(t)$ is the parametrised curve given by

$\displaystyle \alpha(t)$ = $\displaystyle \left(\begin{array}{c}sin^2(t)\\sin(t)cos(t)\end{a rray}\right)$

for $\displaystyle 0 \leq t \leq \pi$.

1. Show that $\displaystyle \alpha(t)$ is parametrised by arc length.
2. Find the length of $\displaystyle \alpha(t)$.
3. Find the normal vector to $\displaystyle \alpha$(t0) at where $\displaystyle 0 < t0 < \pi$.
4. Do you recognise this curve?

I don't have a clue what any of this means, the lecturer rushed through it and I won't have a chance to see her until Tuesday and this is in for Monday. Any help is much appreciated. Thanks.

2. For the first two, show that $\displaystyle |\alpha'|=1$ and calculate $\displaystyle \int |\alpha'|$. The third one is but an easy problem involving analytic geometry only. For the fourth one, call the coordinates x and y,
and calculate an algebraic expression involving them.