Environmental Engineering Reference
In-Depth Information
TABLE 1.1
Vector operations in the Cartesian basis.
A
and
B
are vectors,
λ
is a scalar.
Operation
Notation
Resultant
Negation
−
A
(
−
A
x
)
i
+
(
−
A
y
)
j
+
(
−
A
z
)
k
Addition
A
+
B
(A
x
+
B
x
)
i
+
(A
y
+
B
y
)
j
+
(A
z
+
B
z
)
k
Subtraction
A
−
B
(A
x
−
B
x
)
i
+
(A
y
−
B
y
)
j
+
(A
z
−
B
z
)
k
Scalar (Dot) Product
A
·
B
A
x
B
x
+
A
y
B
y
+
A
z
B
z
Vector (Cross) Product
A
×
B
(A
y
B
z
−
A
z
B
y
)
i
+
(A
z
B
x
−
A
x
B
z
)
j
+
(A
x
B
y
−
A
y
B
x
)
k
Scalar Multiplication
λ
A
λA
x
i
+
λA
y
j
+
λA
z
k
y
axis
q
r
P
j
r
y
i
q
x
x
axis
0
Figure 1.7 2D co-ordinate systems. The Cartesian co-ordinates consist of the set
(x, y)
.
The plane polar co-ordinates consist of the set
(r, θ)
.
system may be used. Two of these are shown in Figure 1.7. The Cartesian 2D
co-ordinate system has basis vectors
i
and
j
and co-ordinates
(x, y)
. The plane-polar
co-ordinates are
(r, θ)
6
where
x
2
tan
−
1
y
r
=
+
y
2
and
θ
=
x
.
(1.7)
The plane-polar system has basis vectors
r
and
ˆ
. These may be expressed in
θ
terms of
i
and
j
as
r
=
i
cos
θ
+
j
sin
θ,
ˆ
θ
=−
i
sin
θ
+
j
cos
θ.
(1.8)
The general position vector in the plane may therefore be written as
r
=
r
r
=
r(
i
cos
θ
+
j
sin
θ)
=
x
i
+
y
j
.
(1.9)
Some care is required when using polar co-ordinates to describe the motion of a
particle since the basis vectors depend on the co-ordinate
θ
, which may itself depend
on time. This means that as the particle moves, the basis vectors change direction.
6
Note the conventional use of
θ
for the angle to the
x
axis rather than
φ
, which is used for the
corresponding angle in the spherical (3D) polar system.
