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1
Review: Vector Analysis
REFERENCES: Arya Chapter 5.
1.1
Vectors
A scalar has only a magnitude. A vector has both a magnitude and a direction. In this
course, vectors will be represented in two ways (bold, or overarrow):
R
=
→
R,
(1)
or in the special case of a unit vector, with a
hat
(e.g.
ˆ
i
). The magnitude of a vector is
represented
R
=

R

=

R

(2)
Vectors can be written in components. For example, in two dimensions:
R
=(
R
x
,R
y
)
(3)
and
R
x
=
R
cos(
θ
)
R
y
=
R
sin(
θ
)
(4)
where
R
=
±
R
2
x
+
R
2
y
tan
θ
=
R
y
R
x
(5)
We also denote vectors with
unit vectors
: For example in 3D cartesian coordinates:
ˆ
i
= (1
,
0
,
0)
,
ˆ
j
= (0
,
1
,
0)
,
ˆ
k
= (0
,
0
,
1)
(6)
where
ˆ
i
is the unit vector associated with the xaxis direction,
ˆ
j
the yaxis direction, and
ˆ
k
the zaxis direction. Each has unit magnitute

ˆ
i

=

ˆ
j

=

ˆ
k

= 1
(7)
A general vector in 3D can then be represented as:
R
R
x
y
z
)=
R
x
ˆ
i
+
R
y
ˆ
j
+
R
z
ˆ
k
(8)
Adding and subtracting vectors can be done componentwise. From this, certain proper
ties follow. Consider vectors
A
,
B
and
C
and scalar
ϕ
:
1.
A
i
+
B
i
=
B
i
+
A
i
(commutative law)
2.
A
i
+(
B
i
+
C
i
) = (
A
i
+
B
i
)+
C
i
(associative law)
3.
ϕ
A
=
B
is a vector
1
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Scalar product
The
scalar product
is defned to be a scalar quantity obtained by multiplying the magnitudes
oF two vectors:
S
=
A
·
B
=
AB
cos
θ
=
±
i
A
i
B
i
(9)
where
θ
is the angle between the vectors. The magnitude oF a vector can be calculated

R

=
√
R
·
R
=
²
±
i
R
2
i
(10)
The scalar product has certain useFul properties:
1.
A
·
B
=
B
·
A
(commutative).
2.
A
·
B
= 0 iF
A
or
B
= 0, or
θ
= 90 degrees (the vectors are
orthogonal
).
3.
A
·
(
B
+
C
)=
A
·
B
+
A
·
C
(distributive)
4. (
ϕ
A
)
·
B
=
ϕ
(
A
·
B
) (associative with scalar multiplication)
example:
Two position vectors are
A
=
ˆ
i
+2
ˆ
j

2
ˆ
k
and
B
=4
ˆ
i
ˆ
j

3
ˆ
k
. ±ind the
magnitude oF the vector From point A to B, the angle
θ
between
A
and
B
, and the component
oF
B
in the direction oF
A
solution:
The vector From point A to B is
B

A
. By components:
B

A
= (4

1)
ˆ
i
+ (2

2)
ˆ
j
+(

2 + 3)
ˆ
k
=3
ˆ
i

ˆ
k
(11)
The magnitude oF this vector is

B

A

=
√
9 + 1 =
√
10
(12)
±rom the scalar product Eq. 9:
cos
θ
=
A
·
B
AB
=
4 + 4 + 6
√
9
√
29
=0
.
8666
(13)
The component oF
B
in the direction oF
A
is
B
cos
θ
, so
B
cos
θ
=
A
·
B
A
=
14
3
.
67
(14)
2
1.3
Vector product
The
vector
(or cross) product is defned to be a vector quantity:
C
=
A
×
B
(15)
i.e.
C
has a magnitude and direction. The magnitude oF
C
is
C
=
AB
sin
θ
(16)
(For 0
< θ < π
), and it’s direction is determined by the righthand rule, defned along the
normal
direction ˆ
n
. Thus, For example, one can defne the area oF a parallelogram with two
sizes
B
and
C
as
area
A
=
B
×
C
=(
BC
sin
θ
)ˆ
n
(17)
The components oF
C
are defned by the relation
C
i
≡
±
j,k
ε
ijk
A
j
B
k
(18)
where
ε
ijk
is the
LeviCivita
symbol, defned:
ε
123
=
ε
231
=
ε
312
= 1
(19)
ε
132
=
ε
213
=
ε
321
=

1
ε
ijk
= 0
otherwise
.
Alternatively, the vector product can be represented by a determinate:
C
=
ˆ
i
ˆ
j
ˆ
k
A
x
A
y
A
z
B
x
B
y
B
z
(20)
in other words
C
x
=
A
y
B
z

A
z
B
y
(21)
C
y
=
A
z
B
x

A
x
B
z
C
z
=
A
x
B
y

A
y
B
x
The vector product has certain useFul properties:
1.
A
×
B
=

B
×
A
(anticommutative).
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This note was uploaded on 09/20/2010 for the course AMATH 261 taught by Professor Rogermelko during the Spring '10 term at Waterloo.
 Spring '10
 RogerMelko

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