Brain Gains (Rapid Recall, Jivin' Generation)

1. For $\vec r(t) = (\cos t, \sin t, t)$, compute $\vec T = \dfrac{d\vec r}{ds}$, $\vec N = \dfrac{d\vec T/dt}{|d\vec T/dt|}$, and $\vec B = \vec T\times \vec N$.

Let's view an animation.

https://www.geogebra.org/m/kaGKS9BB

2. For the circle $\vec r(t) = (3\cos t, 3\sin t)$, compute the curvature $\kappa = \left|\dfrac{d\vec T}{ds}\right|$ and radius of curvature $\dfrac{1}{\kappa}$.

3. Find a vector that orthogonal to the two vectors $(-1,2,5)$ and $(3,0,4)$.

4. Give an equation of the plane that passes through the three points $(2,0,0)$, $(0,3,0)$, $(0,0,5)$.

Group Problems

1. For $\vec r(t) = (4\cos t, 4\sin t, 3t)$, compute $\vec T = \dfrac{d\vec r}{ds}$, $\vec N = \dfrac{d\vec T/dt}{|d\vec T/dt|}$, $\vec B = \vec T\times \vec N$, and $\kappa$.

2. Give an equation of a plane that passes through the points $(2,0,1)$, $(1,-2,0)$, and $(3,3,2)$.

3. For $\vec r(t) = (t^2, 0,t)$, at $t=1$ please compute $\vec T = \dfrac{d\vec r}{ds}$, $\vec N = \dfrac{d\vec T/dt}{|d\vec T/dt|}$, $\vec B = \vec T\times \vec N$, and $\kappa$.