2.3. VECTOR FUNCTIONS 85
2.3.1 CALCULUS WITH VECTOR CURVES
A big advantage of vector functions is that they free us from the tyranny of abstract one-
dimensional position, velocity, and acceleration.
Definition 2.9 If Ev.t/ D .f
1
.t/; f
2
.t/; : : : ; f
n
.t// then the derivative of Ev.t/ is
Ev
0
.t/ D .f
0
1
.t/; f
0
2
.t/; : : : ; f
0
n
.t//:
Here is the payoff. If Es.t/ gives the position of a particle, then its velocity vector is Ev.t/ D
E
s
0
.t/,
and its acceleration vector is Ea.t/ D
E
v
0
.t/. e derivative-based relationships between position,
velocity, and acceleration hold for vector functions just as they do for ordinary functions. e
difference is that we may now describe motion in complex two- and three-dimensional paths.
Example 2.64 Suppose the path of a particle is given by the vector function:
Es.t/ D .sin.t/; cos.2t//
Plot the particle’s path and find its velocity and acceleration vectors.
Solution:
e plot looks like this:
1.0
-1.0
1.0-1.0
Take derivatives to find
Ev.t/ D .cos.t/; 2 sin.2t// and Ea.t / D .sin.t /; 4 cos.2t//
˙
86 2. PARAMETRIC, POLAR, AND VECTOR FUNCTIONS
What do the vector velocity and acceleration mean? ey give (as their magnitude) the amount of
velocity or acceleration, but they also tell us which direction in space the velocity or acceleration
is going.
Example 2.65 If a particle has position Es.t/ D .2 cos.t/; sin.t //, plot the position curve and
show the velocity vectors – starting at the curve – at times t D =6; 2=3; 7=6, and 5=3.
Solution:
Taking a derivative we get that the velocity vector is Ev D .2 sin.t /; cos.t//. So, let’s plot
the curve and the vectors at the specified times.
2.0
t D =6
t D 2=3
t D 7=6
t D 5=3
-2.0
3.0-3.0
Notice that these vectors are tangent to the curve and also show which direction the curve is
oriented. ey show the instantaneous velocity of the particle following the curve.
˙
2.3. VECTOR FUNCTIONS 87
Example 2.66 Find the times when the position curve Es.t/ D .2 sin.t/; cos.t/ C 1/ has
velocity of the greatest magnitude.
Solution:
First, take a derivative to obtain the velocity function:
Ev.t/ D .2 cos.t/; sin.t//
e magnitude of the velocity is then
j.2 cos.t/; sin.t//j D
q
4 cos
2
.t/ Csin
2
.t/ D
p
3 cos
2
.t/ C1
Note that the second step above used the Pythagorean identity, sin
2
.t/ Ccos
2
.t/ D 1. Now
we need to know when this is largest: : : Since 0 cos
2
.t/ 1 it will simply happen when
cos
2
.t/ D 1. is means that cos.t/ D ˙1. So the times are t D k for any whole number k.
˙
Note that the optimization of the function
velocity D
p
3 cos
2
.t/ C1
was not performed by solving a derivative equal to zero. Rather, reasoning based on well know
propeties of the function cosine were used to extract the maximum. Again: mathematics is the
art of avoiding calculation.
PROBLEMS
Problem 2.67 For each possible pair of the following four vectors, say which are orthogonal.
• Ev D .1; 1/ • Er D .1; 2/ • Es D .2; 1/ • Ew D .5; 5/
Problem 2.68 For each possible pair of the following four vectors, say which are orthogonal.
• Ev D .1; 1; 1/ • Er D .2; 2; 0/ • Es D .3; 1; 4/ • Ew D .1; 2; 1/
88 2. PARAMETRIC, POLAR, AND VECTOR FUNCTIONS
Problem 2.69 Plot the following vector curves. You are free to use software.
1. Ev.t/ D .3 sin.t=3/ Ccos.t /; 3 cos.t =2/ Csin.t //
2. Ew.t/ D .3 sin.t=3/ Ccos.2t /; 3 cos.t =2/ Csin.2t //
3. Es.t/ D .2 sin.t=3/ C 2 cos.2t/; 2 cos.t=2/ C 2 sin.2t//
4. Eq.t/ D .sin.t=2/ C 3 cos.3t/; cos.t=2/ C 3 sin.3t//
5. Ea.t / D .3 cos.2t/; 3 sin.t //
6.
E
b.t/ D .t C sin.t/; cos.t //
Problem 2.70 Example 2.64 shows the graph of the position function
Es.t/ D .sin.t/; cos.2t//
and it looks like a part of a parabola. Demonstrate that it is a parabola by finding the Cartesian
form, including the domain on which the curve is defined.
Problem2.71 If Es.t / D .2t; cos.t// is the position of a point, then plot the curve traced by the
point on Œ0; 2 and show the velocity vectors at each multiple of =6 in the interval.
Problem 2.72 For each of the following position vectors, find the velocity and acceleration
vectors.
1. Es.t/ D .3t C 5; 2t C 6/
2. Es.t/ D .t
2
C t C 1; 5 t/
3. Es.t/ D .sin.t/; cos.2t //
4. Es.t/ D .tan
1
.t/; t
2
/
5. Es.t/ D .1 Ct C t
2
; 1 t C t
2
/
6. Es.t/ D .t cos.t/; t sin.t//
Problem 2.73 Find when a particle whose position is given by
Es.t/ D .sin.t C =3/; cos.t//
is traveling parallel to the y-axis.
Problem 2.74 Find a Cartesian function with the same graph as:
Ew.t/ D .3t C 1; 5t C 2/
2.3. VECTOR FUNCTIONS 89
Problem 2.75 Find a Cartesian function with the same graph as
Eu.t/ D .t
2
; 1 3t/
or give a reason it is impossible.
Problem 2.76 Show, by argumentation, that a vector function, all of whose components are
linear functions of t, has the same graph as a Cartesian line.
Problem2.77 e vector function .cos.t/; sin.t /; t/ is a twisting path. First explain what curve
this vector function traces out if we ignore the third coordinate and then do your best to explain
what the shape of the curve is.
Problem 2.78 If f .t/ is a function with domain 1 < t < 1, then describe the vector func-
tion .f .t/; f .t/; f .t// as best you can.
..................Content has been hidden....................
You can't read the all page of ebook, please click here login for view all page.