I am looking for some clarification about these two situations.
1. Find the principal unit normal vector to the curve r(t) = 5ti + 4j when t = 3.
I found this answer to be undefined, as T'(t) = 0 and the magnitude of T'(t) = 0, but I'm not sure.
Theorem Needed: N(t) = T'(t) / ||T'(t)||
2. Find the curvature of the curve r(t) = 2cos(π t)i + 3sin(π t)j (π and t are spaced to avoid πt)
I found the magnitude of r'(t) x r''(t) to be 6π^3, but I'm not sure if the magnitude of r'(t) is 3π. My final answer would come out to be 6/27 if these are correct.
Theorem Needed: K = ||T'(t)|| / ||r'(t)|| = ||r'(t) x r''(t)|| / ( ||r'(t)|| )^3
Any input is appreciated. #1 would be about simplifying sqrt( 4π^2sin^2(π t) + 9π^2cos^2(π t) ) and #2 would be about what T'(t) is and if it end up being undefined.
I am mainly wanting to know if I have the correct answers.