- I-Learn, Class Pictures, Learning Targets, Text Book Practice
- Prep Tasks: Unit 1 - Motion, Unit 2 - Derivatives, Unit 3 - Integration, Unit 4 - Vector Calculus
Brain Gains (Rapid Recall, Jivin' Generation)
1. For $\vec r(t) = (2\cos t, 5t, 2\sin t)$, 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 $\kappa$.
Solution and animation
We have $$\ds\frac{d\vec r}{dt} = (-2\sin t, 5, 2\cos t)$$ which means $\ds\left|\frac{d\vec r}{dt}\right| = \sqrt{29}$. This gives $$\ds\vec T = \frac{ (-2\sin t, 5, 2\cos t) }{\sqrt{29}}.$$ We then compute $$\ds\frac{d\vec T}{dt} = \frac{ (-2\cos t, 0,-2\sin t) }{\sqrt{29}}$$ which means $\ds\left|\frac{d\vec T}{dt}\right| = \frac{2}{\sqrt{29}}$. We then have $$\vec N = (-\cos t, 0,-\sin t).$$ The cross product gives $$\vec B = \frac{ (-5\sin t,-2,5\cos t) }{\sqrt{29}}.$$ The curvature is $$\kappa = \left|\frac{d\vec T}{ds}\right| = \frac{|d\vec T/dt|}{ds/dt} = \frac{2\sqrt{29}}{\sqrt{29}} = \frac{2}{29}.$$
Here's an animation of these topics.
Here's some Mathematica code to compute all the above.
r[t_] := {2 Cos[t], 5 t, 2 Sin[t]}
T[t_] := r'[t]/Norm[r'[t]] // ComplexExpand // Simplify
Nn[t_] := T'[t]/Norm[T'[t]] // ComplexExpand // Simplify
B[t_] := T[t]\[Cross]Nn[t] // ComplexExpand // Simplify
\[Kappa][t] := Norm[T'[t]]/Norm[r'[t]] // ComplexExpand // Simplify
r'[t]
T[t]
T'[t]
Nn[t]
B[t]
\[Kappa][t]
2. Give an equation of the plane that passes through the three points $(2,0,0)$, $(0,3,0)$, $(0,0,5)$.
Solution
Two vectors in the plane are $(-2,3,0)$ and $(-2,0,5)$ (found by subtracting the first point from each of the other two). The cross product of these two vectors is $(15,10,6)$, and this vector is orthogonal to any vector in the plane.
Let $(x,y,z)$ be any point in the plane. Then the vector $(x-2,y-0,z-0)$, a vector from $(2,0,0)$ to $(x,y,z)$, is a vector in the plane and as such must orthogonal to $(15,10,6)$. This gives an equation of the plane as $$(15,10,6)\cdot (x-2,y-0,z-0)=0.$$ We'll simplify this a bit (removing the dot product) to obtain $$15(x-2)+10(y-0)+6(z-0)=0.$$
Group Problems
1. For $\vec r(t) = (3t, 2\sin t, 2\cos t)$, 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 $\kappa = \left|\dfrac{d\vec T}{ds}\right|$.
2. A plane passes through the points $P=(2,0,1)$, $Q=(1,-2,0)$, and $R=(3,3,2)$. Give a nonzero vector $\vec n$ that is orthogonal to both $\vec {PQ}$ and $\vec{PR}$, which we call a normal vector to the plane (it will be orthogonal to every vector in the plane). Then let $S = (x,y,z)$ be another point in the plane. Use that fact that $\vec PS$ is a vector in the plane, and $\vec n$ is a normal vector to the plane, to give an equation of the plane.
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 $\kappa = \left|\dfrac{d\vec T}{ds}\right|$. The computations, by hand, will get quite ugly very quickly. Once you've computed $d\vec T/dt$, you don't need any more derivatives, which means at that point you can plug in $t=1$ and then simplify before continuing on to get $\vec N$, $\vec B$, and $\kappa$.
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