The theorem you have is for a curve. This problem gives you a line. The "unit normal" is a vector perpendicular to the line, of length 1. By convention, it is the normal in the "positive" rotation. For this line, which is a horizontal line, that would be j.

. Its length is , not .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.