C Roettger, Fall 14
Math 265 Quiz 4C – Solutions
Problem 1 A particle is traveling on a curve with position at time t given
as
r(t) = 5 sin(t)i + 3tj.
a) Find the velocity and acceleration vectors.
b) Find the unit tangent vector T and the unit normal vector N at time
t = π/2.
Solution. For a),
v = 5 cos(t)i + 3j
a = −5 sin(t)i
For b), T = j for t = π/2 and N = −i. We know that N is orthogonal to
N, so it is a multiple of i. It has length 1, so it is ±i. Finally, it points to
the same side of the tangent line as the curve, so it has to be −i (see graph
below). Alternatively, N points to the same side of the tangent line as a.
This is much easier than computing dT /dt.