Question

Let C be a smooth curve with parametrization r(t). We know from class that at time t its acceleration vector has the form

a(t) = aT(t)T(t) + aN(t)N(t),
where aT(t) and aN(t) are the tangential and normal components of the acceleration. Use this fact to show that, for each t, the vector r0(t)×r00(t) is parallel to B(t).

Answer 1

Using the given fact to show that, for each t, the vector r'(t)×r"(t) is parallel to B(t), below is proof.

B(t) = λ [r'(t) * r"(t)] where, λ = 1 / [ aₙ(t) ||r'(t)|| ]

Explanation:

a(t) = aT(t)T(t) + aN(t)N(t), where aT(t) and aN(t) are the tangential and normal components of the acceleration.

In the attached workings, the above fact is used to show that for every 't', the vector r'(t)×r"(t) is parallel to B(t).