Finding the point of intersection between two curves. (Vectors)

[Jaqsan]

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]

Welcome to PF Jaqsan,

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. :-(

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)?

 

[Jaqsan]

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.

 

[Jaqsan]

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)?

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.

 

[haruspex]

s=8
t=1

What is the vector r1 when t=1?

 

[Dick]

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.

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]

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.

Okay, so I got
r1'(1) = <1,-1,2>
r2'(8)= <-1,1,16>
So do I dot them to get them in one (x,y,z) form?

 

[Dick]

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?

Ummm, to answer the (x,y,z)=___ you just substitute. To answer the angle question you want to form a dot product.

 

[Jaqsan]

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.

Ummm, to answer the (x,y,z)=___ you just substitute. To answer the angle question you want to form a dot product.

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]

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>

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]

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.

Thanks. I figured it out. I was just being retarded. My answers are (1,3,64) and 40degrees

 

[cepheid]

Thanks. I figured it out. I was just being retarded. My answers are (1,3,36) and 40degrees

I think you mean (1,3,64), right?

 

[Jaqsan]

I think you mean (1,3,64), right?

My bad. Exhibiting my retardness once again. (1,3,64) and 40 degrees.