Finding a tangent line to a position vector?

Question

The position vector for a partivle moving on a helix is r(t) =

Find an equation for a tangent line to r(t) =(4pi)/w

What i have done so far is took the derivative of r(t):

r'(t)=<-wsinwt, wcoswt, 2t>

Then I plugged in t = (4pi)/w

and got <0,4pi/w, 8pi/w>

Is this right, am I going in the right direction? Also, how do you get an equation from this?

Answer 1

you are right.
r(4pi/w)=<1,0,16pi^2/w^2>
so the parametric equations of the line are:

x= 1
y=(4pi/w) t
z=16pi^2/w^2 + (8pi/w) t

equivalently

=<1,0,16pi^2/w^2>+t<0,4pi/w, 8pi/w>

just keep in mind that here t is the parameter of the tangent line.
if this is confusing you could use another letter, like s or u, etc

Answer 2

This is the right direction.

The equation of the line can be written in the form

v(s) = r(t0) + s*r'(t0)

in this case, t0=4pi/w, and s is a parameter. The line consists of all points of this form, where s is any real number.