Velocity Vector Versus Tangent Line

[lovelylila]

I have encountered a problem in my Calculus homework.

I have a position function, r(t)= (t^2)i + (4t)j and in my homework, I am asked to find the tangent to this curve at the point t=3. I did this by finding dy/dx, or 2t/4 @ t=3 is 6/4. However, I am also asked to relate this to the velocity vector for the position function @ t=3, but I don't understand the relationship. Would they share the same slope? Any help is very much appreciated! :-)

[jhicks]

Would they share the same slope? Any help is very much appreciated! :-)

Well you have the position's function. The tangent line necessarily has a slope of dr/dt, which is equivalent to the velocity so yes.

[lovelylila]

Oh that makes sense! Thank you very much :-) But they're not the same line...are they?

[jhicks]

The velocity has no inherent sense of position, so you can't really compare velocity and a tangent line even at the same time t. A tangent line more or less answers the question "where would you be at some time you knew you were at position r(t) at time t and maintained a constant velocity for all time?", but don't read too far into this.