Unit Vectors
Goal:
It is the goal of this simulation to examine and
clarify some key ideas behind the unit vectors 
and 
.
Page 29 of K&K illustrates how the unit vectors 
,
,
,
and
change after a particle moves from position 
1
to 2,
but this simulation allows you to watch how the unit vectors change with
time. By watching this simulation, you should be able to
1. Understand what K&K means by “We can think of
and
as being functionally dependent on θ” (pg. 29)
2. See how
and
are different from
and
Basic Premise:
A robotic ladybug wishes (yes, it has desires) to spread its visible antennae. The programmers of this robotic ladybug have designed it so that initially, both antennae are at rest horizontally. At time t = 0 sec, the antennae both move simultaneously with a constant angular velocity, ω. At an angle of θ1 from the horizontal, one of the antennae stops moving and the other continues moving with the same angular velocity
ω. Eventually, when the second antenna makes an angle of θ2 with the horizontal, it stops. Let us examine the motion of the moving antenna and the translation of the unit vectors associated with it!
Analysis:
When the first antenna stops at θ1 ,
points out radially and itself has an angle of θ1 from
the horizontal.
is perpendicular to
and has an angle θ1 from the vertical.
and
point along the increasing x and
increasing y directions, respectively. When we follow the path of the
second antenna and look at the unit vectors at θ2,
we see that
and
look exactly the same, whereas the angles of
and
change as θ2 changes. Thus, and
are dependent on the angle θ while
and
are fixed.
For further exploration, see simulation ejs_Polars3.RadialCompOnly.