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

1. Feb 10, 2013

### Jaqsan

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

3. 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. :-(

2. Feb 10, 2013

### cepheid

Staff Emeritus
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)?

3. Feb 10, 2013

### 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.

4. Feb 10, 2013

### 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.

5. Feb 10, 2013

### haruspex

What is the vector r1 when t=1?

6. Feb 10, 2013

### Dick

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.

7. Feb 10, 2013

### Jaqsan

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?

8. Feb 10, 2013

### Dick

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

9. Feb 10, 2013

### Jaqsan

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>

10. Feb 10, 2013

### Dick

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.

Last edited: Feb 10, 2013
11. Feb 10, 2013

### Jaqsan

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

Last edited: Feb 10, 2013
12. Feb 10, 2013

### cepheid

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

13. Feb 10, 2013

### Jaqsan

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

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