C Roettger, Fall 14 Math 265 Quiz 4D – Solutions Problem 1 A particle is traveling on a curve with position at time t given as r(t) = t3 i + t2 j. a) Find the velocity and acceleration vectors. b) Find the unit tangent vector T and the unit normal vector N at time t = 2. Solution. For a), v = 3t2 j + 2tj a = 6ti + 2j For b), 1 1 (12i + 4j) = √ (3i + j). 160 10 when t = 2. Since N is orthogonal to T and ||N|| = 1, T= √ 1 N = ± √ (i − 3j). 10 Finally, N points to the same side of the tangent line as the curve, so the plus sign is correct in the above two possibilities for N (see graph below). Alternatively, N points to the same side of the tangent line as a, and N.a > 0. This is much easier than computing dT /dt.