Task 10.1

Given a curve with parametrization $\vec r(t)$, we have already seen that a unit tangent vector is given by $\ds \vec T = \frac{d\vec r}{ds} = \frac{d\vec r/dt}{|d\vec r/dt|}$. Note that this vector has constant length of 1, which means that it's derivative, so $\frac{d\vec T}{dt}$, is orthogonal to $\vec T$. This vector describes how the direction of motion changes. The vector $\ds\vec N = \frac{d\vec T/dt}{|d\vec T/dt|}$ provides a unit vector, we call the principle unit normal vector, that describes the direction in which an object is turning. The cross product $\vec B = \vec T\times \vec N$ we call the binormal vector. These three vectors, namely $\vec T$, $\vec N$, and $\vec B$, provide what we call the Frenet or TNB frame, and are commonly used when describing motion.

  1. For the curve $\vec r(t) = (3\cos t, 3\sin t, 4t)$, compute $\vec T$, $\vec N$, and $\vec B$. Show how you obtained each step in your computations.
    • The definitions of $\vec T$ and $\vec N$ both made them one unit long. How long is $\vec B$?
  2. For the curve $\vec r(t) = (t,0,t^2)$, compute $\vec T$, $\vec N$, and $\vec B$. Show how you obtained each step in your computations. If things get ugly quite quickly, because of a quotient rule, then you're on the right path.

For a visual representation of the Frenet Frame, please visit this Geogebra site. It's possible to create a very similar visual in Mathematica or Python (something you could aim for with a self-directed learning project).

Task 10.2

Given a parametric curve with parametrization $\vec r(t)$, the curvature vector is the rate of change of the direction of motion with respect to arc length, so $\ds \vec \kappa = \frac{d\vec T}{ds}$. We compute the derivative with respect to arc length so that we obtain a physical property of the curve, rather than a property that relates to how quickly we traverse the curve. The curvature is the magnitude of the curvature vectors, so $$\kappa = |\vec \kappa| = \left| \frac{d\vec T}{ds} \right|.$$ The radius of curvature is the quantity $1/\kappa$.

  1. Explain why $\ds \kappa = \frac{|d\vec T/dt|}{|d\vec r/dt|}$.
  2. For the circle $\vec r(t) = (5\cos(2t), 5\sin(2t))$, compute the curvature $\kappa$ and radius of curvature $1/\kappa$.
  3. For the helix $\vec r(t) = (3\cos(t), 3\sin(t),4t)$, compute the curvature and radius of curvature.
  4. For the parabola $\vec r(t) = (t,t^2)$, at $t=0$ compute the curvature $\kappa(0)$ and radius of curvature.
  5. Draw the parabola from the previous part. How would you interpret the radius of curvature at $t=0$ in this context?

Task 10.3

Let $P=(a,b,c)$ be a point on a plane in 3D. Let $\vec n=(A,B,C)$ be a normal vector to the plane (so the angle between the plane and $\vec n$ is 90$^\circ$). Let $Q=(x,y,z)$ be another point on the plane.

  1. What is the angle between $\vec {PQ} = (x-a,y-b,z-c)$ and $\vec n=(A,B,C)$?
  2. Explain why an equation of the plane through $P$ with normal vector $\vec n$ is $$A(x-a)+B(y-b)+C(z-c)=0.$$
  3. Consider the three points $R=(1,0,0)$, $S=(2,0,-1)$, and $T=(0,1,3)$. Give an equation of the plane which passes through these three points. [You already have a point on the plane. With three points, you can get two vectors that are in the plane. How can you get a vector that is normal to the plane?]

Task 10.4

The last problem for prep each day will point to relevant problems from OpenStax. Spend 30 minutes working on problems from the sections below.

  • Return to any of the previous day's OpenStax problems to locate extra practice.