Determine the parametric equations of the tangent line to the curve r(t) =< cos 2t; sin 3t; sin 4t > when t = pi/4

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Determine the parametric equations of the tangent line to the curve r(t) =< cos 2t; sin 3t; sin 4t > when t = pi/4

Mathematics
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we want the time derivative of r(t) r'(t)=<-2sin(2t), 3cos(3t), 4cos(4t)> now find r'(pi/4) r'(pi/4)=<-2, -3sqrt(2)/2, -4> so we have a direction vector, now we want a point on the line. We can use r(pi/4) r(pi/4)=<0, sqrt(2)/2, 0> so the equation of the tangent line is T=<-2s, sqrt(2)/2-(3sqrt(2)/2)*s, -4s> where s is an arbitrary parameter.
I took the derivative but was unsure where to go from there, thanks!
in the last equation why the variables swapped?

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so it's not confused with t, since it doesn't need to be the same variable.

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