Question

Find a unit speed parametrization of α(t) = (r cost, r sin t, 0) for t ∈ R.

Answer 1

To find a unit speed parametrization of α(t) = (r cos(t), r sin(t), 0) for t ∈ R, we need to determine the derivative of the position vector with respect to time and normalize it to unit speed.

Step-by-step explanation:

To find a unit speed parametrization of α(t) = (r cos(t), r sin(t), 0) for t ∈ R, we need to determine the derivative of the position vector with respect to time.

The position vector is given as α(t) = (r cost, r sin t, 0).

Let's find the derivative of α(t) with respect to t.

We have α'(t) = (-r sin t, r cost, 0).

Now, let's find the magnitude of α'(t) to normalize it to unit speed.

The magnitude of α'(t) is √((-r sin t)^2 + (r cos t)^2 + 0^2) = √(r^2 sin^2 t + r^2 cos^2 t) = √(r^2 (sin^2 t + cos^2 t)) = √(r^2) = r.

Therefore, a unit speed parametrization of α(t) is given by α(t) = (r cos(t), r sin(t), 0) for t ∈ R.