Intersection of two vector-valued functions

[karens]

Homework Statement

Prove that r1 and r2 intersect at (1,1,3).
Let r1 and r2 be defined as:

r1(t)=t^2i+tj+3t^3k
r2(t)=(t−1)i+(1/4)t^2j+(5−t)k

Homework Equations

Intersection is derived from r1=r2.

The Attempt at a Solution

I only formally get to this point and then start messing up. It gets messy and I know I'm not doing the right thing.

r1=r2 --> t^2i+tj+3t^3k = (t−1)i+(1/4)t^2j+(5−t)k

 

[lanedance]

it may help to write the curves with different parameterisation variables
r1(s)=s^2i+sj+st^3k
r2(t)=(t−1)i+(1/4)t^2j+(5−t)k

then solve for correspdoning intersection point in terms of s & t

 

[Mark44]

Homework Statement

Prove that r1 and r2 intersect at (1,1,3).
Let r1 and r2 be defined as:

r1(t)=t^2i+tj+3t^3k
r2(t)=(t−1)i+(1/4)t^2j+(5−t)k

Homework Equations

Intersection is derived from r1=r2.

The Attempt at a Solution

I only formally get to this point and then start messing up. It gets messy and I know I'm not doing the right thing.

r1=r2 --> t^2i+tj+3t^3k = (t−1)i+(1/4)t^2j+(5−t)k

Use a different parameter for each vector function. Just because the two curves intersection, there's no guarantee that the same value of the parameter works in both functions.

In this problem, all you need to do is to show that (1, 1, 3) is a point on both curves. In the first function, what value of t gives this point? In the second function what value of s gives the same point?