Question

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

Answer 1

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.

Answer 2

I took the derivative but was unsure where to go from there, thanks!

Answer 3

in the last equation why the variables swapped?

Answer 4

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