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.