The TNB Frame

In general, calculating $\vN$ as a function of the parameter $t$ ends up being very difficult because there are multiple compositions of functions involved which means there is a nesting of chain rules involved in the definition. In this activity, we will examine how to go through the direct calculation of $\vN$ for a curve parameterized by $\vr(t) =\langle t,t^2,t^3\rangle$.

1. Calculate velocity $\vv(t)$ and $\text{speed}(t)$ for the parameterization $\vr(t) =\langle t,t^2,t^3\rangle$.

2. Now calcula…

1. Each race car has a 1 meter arrow attached to the hood of the car which points straight ahead. This arrow will measure the three dimensional vector that is the direction of travel. Is the direction of travel a property of the driver or the road? Be sure to explain your answer.

2. The turn signal is measured by checking to see if the steering wheel is turned left or right. We are not measuring how much…

Consider the moving particle whose position at time $t$ in seconds is given by the vector-valued function $\vr$ defined by $\vr(t) = 5t \vi + 4\sin(3t) \vj + 4\cos(3t) \vk$. Use this function to answer each of the following questions.

1. Find the unit tangent vector, $\vT(t)$, to the space curve traced by $\vr(t)$ at time $t$. Write one sentence that explains what $\vT(t)$ tells us about the particle's motion.

2. Determine the speed of the particle moving along the space curve wit…