The problem involves finding the point of intersection between two parametric curves defined by r1(t) = (t, 4-t, 63+t^2) and r2(s) = (9-s, s-5, s^2). Additionally, the task includes determining the angle of intersection between the curves.
Some participants have provided guidance on finding tangent vectors and using the dot product to calculate the angle between them. There is ongoing exploration of the intersection point and the relationship between the parameters t and s.
Participants express uncertainty about the steps following the substitution into the parametric equations and the interpretation of the results. There is mention of potential confusion regarding the values obtained for the intersection point and angle.
Jaqsan said:Homework Statement
At what point do the curves r1(t) = (t, 4-t, 63+t^2) and r2(s)= (9-s, s-5, s^2) intersect?
Answer in the form: (x,y,z) = ____
Find the angle of intersection theta to the nearest degree.
Homework Equations
The Attempt at a Solution
i: t=9-s
j: 4-t=s-5
k: 63+t^2=s^2
i/j: t-9s
k: 63+(9-s)^2=s^2
"Solving for "s""
s=8
t=1
...
I know not what to do from here. :-(
cepheid said:Welcome to PF Jaqsan,
It sounds like you want to find the "direction" of each curve at the point of intersection. Can you think of a vector that describes this, and how to compute that vector given r(t)?
What is the vector r1 when t=1?Jaqsan said:s=8
t=1
Jaqsan said:I don't understand. The question asks for the point of intersection of the two curves. Do I find the derivative, dot product? Just point me in the right direction please.
Dick said:You already found the intersection point correctly. So the two tangent vectors are r1'(1) and r2'(8). What are they? Then the angle between two vectors a and b is a.b/(|a||b|). Use that.
Jaqsan said:Okay, so I got
r1' = <1,-1,2>
r2' = <-1,1,16>
So do I dot them to get them in one (x,y,z) form?
Dick said:You already found the intersection point correctly. So the two tangent vectors are r1'(1) and r2'(8). What are they? Then the angle between two vectors a and b is a.b/(|a||b|). Use that.
Dick said:Ummm, to answer the (x,y,z)=___ you just substitute. To answer the angle question you want to form a dot product.
Jaqsan said:I thought I already did the substitution. r1' = <1,-1,2> r2' = <-1,1,16> What I'm trying is it looks like there are two numbers for each value <x,y,z>
Dick said:You substituted correctly into the derivatives. That's fine. When they are asking for the intersection point (x,y,z) you should substitute t=1 into r1(t) or s=8 into r2(s). That's what you calculated for the intersection point, yes? They had better both be the same.
Jaqsan said:Thanks. I figured it out. I was just being retarded. My answers are (1,3,36) and 40degrees
cepheid said:I think you mean (1,3,64), right?