C Roettger, Fall 14
Math 265 Quiz 3B – Solutions
Problem 1 A particle moves along a curve with position
r(t) = 3ti + 2j + t3 k
a) Find the velocity vector v(t) of the particle.
b) Are there any time(s) t when the velocity vector is parallel to the line
below? if so, which times?
x = 2 + 4t,
y = 3,
z =5+t
Solution. a) First,
v = (3, 0, 3t2 ).
b) The velocity vector needs to be a multiple of the direction vector w =
(4, 0, 1) for the line. As a vector equation, we are looking for times t when
there exists a scalar c such that
(3, 0, 3t2 ) = c(4, 0, 1)
which gives 3 = 4c and 3t2 = c. Substitute eg the second equation into the
first to get
3 = 12t2
so we get the times t = ±1/2, and c = 3/4.