C Roettger, Fall 14
Math 265 Quiz 3C – Solutions
Problem 1 A particle moves along a curve with position
r(t) = 3i + 4tj + t4 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,
y = 3 − t,
z =5+t
Solution. a) First,
v = (0, 4, 4t3 ).
b) The velocity vector needs to be a multiple of the direction vector w =
(0, −1, 1) for the line. As a vector equation, we are looking for times t when
there exists a scalar c such that
(0, 4, 4t3 ) = c(0, −1, 1)
which gives 4 = −c and 4t3 = c. Substitute eg the second equation into the
first to get
4 = −4t3
so we get the time t = −1, and c = −4.