Need help with this vector problem -- Thank you

[uchuu-man chi]

Homework Statement

Let L₁ be the tangent line to r(t) at the point t = a and let L₂ be the tangent line where t = b. Find the equation of the lines L₁. Find the equation of the lines L₁ and L₂ and find the points of intersection.

r(t) = <f(t), g(t), h(t)>

*bolded letters are vectors

Homework Equations

The Attempt at a Solution

I just wanted to tell you guys my thought process and would you correct me wherever I am wrong. The question has more to it, but the computation would be menial. I'm just having trouble with what vectors to use and all that stuff.

Steps I would take:
-r(a) would give me a point on L₁ and r(b) would give me a point on L₂.
-find T(t)
-find T(a) and T(b)
-for L₁, the equation of line would be (f(a), g(a), h(a)) + T(a)t
-same step for L₂ as L₁
-find if there is an intersection by setting the parameters of x,y, and z of the lines equal

Am I using the correct vector, T(t), for the second step?

 

[BvU]

Did you forget to tell us what T is ?
And: do you always expect to find an intersection ?

 

[uchuu-man chi]

Did you forget to tell us what T is ?
And: do you always expect to find an intersection ?

OOps sorry

the T(t) would be $$\frac {\vec r '(t)} {||\vec r '(t)||}$$
And no I wouldn't expect to always find an intersection. The lines could be parallel or skew

 

[BvU]

The question has more to it

In that case: so far, so good !