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?
To find the point of intersection between two vectors, you need to set up a system of equations with the x, y, and z components of each vector. Then, solve for the variables to find the coordinates of the point of intersection.
Yes, you can find the point of intersection between two curves using algebraic methods. This involves setting the equations of the curves equal to each other and solving for the variables.
The point of intersection between two curves represents the coordinates of a shared point on both curves. This point can be used to solve various problems in geometry, physics, and engineering.
Yes, it is possible for two curves to have more than one point of intersection. This occurs when the curves intersect at multiple points along their paths.
Yes, the point of intersection between two curves can be imaginary. This occurs when the curves do not intersect in real space, but their equations have a common solution in imaginary numbers.