Question #117667
Let E be a Euclidian vector space, let (i,j) be its standard basis. Note C
the circle centered at the origin with radius a, where a is a real positive number. ⃗⃗
1. Let δ = (a cos as )i+(a sin as )j. Show that ([0, 2πa], δ) is a unit speed parametrization of C
1
Expert's answer
2020-05-24T15:08:52-0400

First, the equation of a circle centered at the origin p.O(0;0)p.O\left(0;0\right) and radius R=a>0R=a>0 has the form



x2+y2=a2x^2+y^2=a^2

For example, a=3a=3





Secondly, from physics we know the relationship between the velocity vector and the radius vector



r=x(t)i+y(t)jv=drdt=dx(t)dti+dy(t)dtjv=vx(t)i+vy(t)jv=vx2+vy2\overrightarrow{r}=x(t)\cdot\overrightarrow{i}+y(t)\cdot\overrightarrow{j}\longrightarrow\\[0.3cm] \overrightarrow{v}=\frac{d\overrightarrow{r}}{dt}=\frac{dx(t)}{dt}\cdot\overrightarrow{i}+\frac{dy(t)}{dt}\cdot\overrightarrow{j}\\[0.3cm] \overrightarrow{v}=v_x(t)\cdot\overrightarrow{i}+v_y(t)\cdot\overrightarrow{j}\longrightarrow\\[0.3cm] \left|\overrightarrow{v}\right|=\sqrt{v_x^2+v_y^2}

In our case, the natural variable tt- time was replaced by the general variable ss .


vδ(s)=(acosas)δx(s)i+(asinas)δy(s)iδ(s)=δx2(s)+δy2(s)=(acosas)2+(asinas)2==a2(cos2as+sin2as)=a1=a\overrightarrow{v}\equiv\overrightarrow{\delta}(s)=\underbrace{\left(a\cos as\right)}_{\delta_x(s)}\cdot\overrightarrow{i}+\underbrace{\left(a\sin as\right)}_{\delta_y(s)}\cdot\overrightarrow{i}\longrightarrow\\[0.3cm] \left|\overrightarrow{\delta}(s)\right|=\sqrt{\delta_x^2(s)+\delta_y^2(s)}=\sqrt{\left(a\cos as\right)^2+\left(a\sin as\right)^2}=\\[0.3cm] =\sqrt{a^2\left(\cos^2as+\sin^2as\right)}=a\cdot\sqrt{1}=a

Conclusion,



δ(s)={is a unit speed,ifa=1is not a unit speed,ifa1\left|\overrightarrow{\delta(s)}\right|=\left\{\begin{array}{l} \text{is a unit speed},\quad\text{if}\quad a=1\\[0.3cm] \text{is not a unit speed},\quad\text{if}\quad a\neq1 \end{array}\right.


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