ECE 329
Homework 1
Due: Jan 27, 2009, 5PM
1.
Review exercises on vectors:
Consider the 3D vectors
A
= 3ˆ
x
+ˆ
y

2ˆ
z,
B
=ˆ
x
y

ˆ
C
x

y
+ 3ˆ
where
ˆ
x
≡
(1
,
0
,
0)
,
ˆ
y
≡
(0
,
1
,
0)
, and
ˆ
z
≡
(0
,
0
,
1)
constitute an orthogonal set of unit vectors (denoted
as
a
x
,
a
y
, and
a
z
in the text, respectively) directed along the principal axes of a righthanded Cartesian
coordinate system.
Vectors can also be represented in component form — e.g.,
A
= (3
,
1
,

2)
, which is equivalent to
3ˆ
x
y

z
.
Determine:
a) The vector
D
≡
A
+
B
,
b) The vector
A
+
B

4
C
,
c) The vector
magnitude

A
+
B

4
C

.
d) The unit vector
ˆ
u
along vector
A
+2
B

C
.
e) The
dot product
A
·
B
.
f) The
cross product
B
×
C
.
g) The
scalar triple product
A
·
(
B
×
C
)
.
h) If vectors
A
,
B
,
C
above are measured in units of meters, what would be the unit of the triple
product
A
·
(
B
×
C
)
and what would be the geometrical interpretation of

A
·
(
B
×
C
)

?
2.
Volume integral review:
Let
ρ
(
x,y,z
)=
x
2
+
y
2
+
z
2
C/m
3
denote the electrical
charge density
function — i.e., charge per
unit volume — in a region of space represented in Cartesian coordinates.