Calc Problem: Find Angle of Intersection Between r1(t) & r2(t) at Origin

  • Context: Undergrad 
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Discussion Overview

The discussion revolves around finding the angle of intersection between two parametric curves, r1(t) and r2(t), at the origin. Participants explore the problem's requirements and methods for calculating the angle, with a focus on derivatives and tangent lines.

Discussion Character

  • Mathematical reasoning, Debate/contested, Homework-related

Main Points Raised

  • One participant states the problem and provides the parametric equations for the curves r1(t) and r2(t).
  • Another participant identifies that the curves intersect at the origin when t=0 and attempts to find the angle using derivatives, leading to a calculation of the angle between the tangent vectors.
  • A third participant questions the interpretation of the hint provided in the problem, suggesting that it was meant to guide towards considering tangent lines.
  • Another participant interprets the hint differently, believing it suggests avoiding tangents altogether.
  • A later reply indicates agreement with the previous angle calculation, suggesting that both participants arrived at the same result.

Areas of Agreement / Disagreement

Participants express differing interpretations of the hint regarding the use of tangent lines, leading to a lack of consensus on the approach to solving the problem. However, there is agreement on the calculated angle of intersection.

Contextual Notes

Participants have not fully resolved the implications of the hint, nor have they clarified the assumptions underlying their calculations of the angle between the curves.

Juntao
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The curves r1(t)=<t,t^2,t^3> and r2(t)=<sin t, sin 2t, t> intersect at the origin. Find their angle (acute) of interesection correct to the nearest degree. (Think! What angle are you trying to locate? Now dn't go off on a tangent.)

So that's the problem.
All I got so far is
r1(t)=t(i)+t^2(j)+t^3(k)
r2(t)=sint(i)+sin 2t(j)+t(k)

Now I'm stuck. I'm not sure where to go now.
 
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They intersect at the origin at t=0. Now, I only know how to find the angle between two straight lines, so I would disregard the hint and take the derivative. You get r1'(t)=<1,2t,3t^2>=<1,0,0> and r2'(t)=<cost,2cos2t,1>=<1,2,1>.
Now how do you find the angle f between two vectors?
A*B=ABcosf
1=(1)sqrt(6)cosf
f=arccos(sqrt(6)/6)=66 degrees
 
How is that "disregarding" the hint? The purpose of the hint was to direct you to the tangent lines of the curves.
 
I interpreted, "Don't go off on a tangent" to mean "Tangents aren't the way to solve the problem."
 
Great. My answer matched yours.
 

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