Acute Angle between two vector-valued functions

[SolfegeDuck]

1. The problem statement, all variables and given/known data
Quick question. I need to find cos \vartheta, where \vartheta is the angle (acute) between two vector-valued functions (which are tangent lines).


2. Relevant equations

I think this is relevant:

cos\vartheta = (u dot v)/(\left\|u|| * \left\|v||)

3. The attempt at a solution

Is that the correct equation to use? If so, well, the two equations I'm using both have "t" in them, since they're VVFs. But what do I put in for t? I get the feeling that it's blatantly obvious...Thanks for helping!

[zcd]

Yes, that's the right equation. Dot product is defined as multiplying i, j, and k components of each vector (assuming 3 component vectors) to return a scalar value. |u||v| means multiplying the scalar magnitudes of vector u and v.

[SolfegeDuck]

Great, thanks! But when I actually do out the equation, I have all of those "t" variables everywhere (2t+1) - (3t^3), etcetera...what do I substitute in for that?

[zcd]

That depends on the question. Either it gives you a specific instance for when the two vectors cross or it wants the angle in general form where t is ambiguous.

[SolfegeDuck]

The point is (1,1,3). But for u, t = 3, and for v, t = 4. So, do I use one of those t-variables, or go with something else entirely?

[HallsofIvy]

It shouldn't be hard to see that you need to use both of them- one applies to u and the other to v. In fact, instead of using t in both functions, it would be better to use, say, t1 for u and t2 for v. Then set t1= 3, t2= 4.