Fall’2016 Semester
METR 3113 – Atmospheric Dynamics I: Introduction to Atmospheric Kinematics and Dynamics
Lecture 9. September 12, 2016
Topics: Scalars and vectors. Notation. Vector addition. Vector subtraction. Unit vector. Scalar (dot) product.
Unit vectors in Cartesian coordinates. Vector magnitude. Vector projection on a coordinate axis (vector
component). Multiplication of vector by a scalar. Examples of operations with vectors.
Reading: Section 1.1 of Holton and Hakim, Section 3 of Fiedler.
1. Scalars and vectors
A physical quantity (or variable) that has magnitude only is called a scalar (see Class 2). Examples: mass m,
temperature T, pressure p, density ρ, wind speed V, gas constant of the dry air Rd. Scalars can be constant or
functions of space and time.
A physical quantity (or variable) that has magnitude and direction is called a vector (see Class 2). It can be
represented in space (or on a plane) by an arrow. An arrow (over-arrow to be more specific) is often used also to
denote a vector quantity as b or  . In this class, we will be denoting vectors either in this manner or using a
bold symbol, i.e., like b or Ω . Examples of vector quantities: acceleration a (or a), velocity u (or u), gradient
of a scalar field, e.g., of temperature, T (the meaning of the gradient as a vector operation will be discussed
during next classes), Earth’s angular velocity  (or Ω ).
2. Vector addition
Consider two vectors, c and d :
To add d to c , place the tail of the second vector to the tip of the first. The sum of two vectors, vector c  d , is
the vector that extends from the tail of c to the tip of d . The same vector results from by adding c to d , so
c  d  d  c , which indicates that vector addition is commutative (see illustration below).
Vector addition is also associative. This property is illustrated below, where a  b  c  (a  b)  c  a  (b  c) .
3. Vector subtraction
This operation is closely related to the vector addition: to subtract vector d from vector c add negative of
vector to vector c (see illustration below), that is
c  d  c  ( d) .
4. Unit vector
A unit vector is defined as a dimensionless vector that has a magnitude of 1. The magnitude of the vector b (in
Cartesian coordinates it would be a measure of its length, see p. 6) is written as b  b . The unit vector b̂ (note
the hat) in the direction of vector b is obtained through dividing b by its magnitude:
b̂ 
b
b

b
,
b
which also means that vector b may be written as a product of its magnitude and associated unit vector:
b  bbˆ  b bˆ .
5. Scalar/dot product of two vectors
Consider two vectors, a and b with angle  between them (chosen as the smallest of the two angles).
The scalar (dot) product of a and b is defined as the following scalar quantity:
a  b  a b cos   ab cos  .
The scalar product is commutative:
a  b  b  a  b a cos   ba cos  ,
and distributive:
a  (b  c)  a  b  a  c ,
From the definition of the scalar product it follows that for parallel vectors a and b (with   0 ) its value is
2
just a b  ab , and that the scalar product of a vector by itself, b  b  b  b2 is equal to the square of the
vector magnitude, the positively defined quantity.
Zeroness of the dot product, a  b  0 may be a result of a  0 , b  0 (or both vectors being zero vectors), or
vectors being perpendicular to each other (in the latter case cos   0 ).
6. Vectors in the Cartesian coordinate system
Consider a right-handed Cartesian coordinate system (X, Y, Z) with unit vectors ˆi, ˆj, kˆ (constituting the socalled orthogonal basis) directed along the respective coordinates (see illustration above). From correspondence
of the unit vector directions to the Cartesian coordinate directions, it follows that vectors ˆi, ˆj, kˆ are
perpendicular to each other, i. e., in terms of the scalar product:
ˆi  ˆi  1, ˆi  ˆj  0, ˆi  kˆ  0,
ˆj  ˆj  1, ˆj  ˆi  0, ˆj  kˆ  0,
kˆ  kˆ  1, kˆ  ˆi  0, kˆ  ˆj  0. .
The projection of a vector c on a particular coordinate direction (also called the component of a vector in this
direction) is defined as the dot (scalar) product of c with the unite vector in that particular direction. The
projection/component is therefore a scalar. In 3-D Cartesian coordinate system, vector c has therefore the
following components:
cx  c  ˆi, c y  c  ˆj, cz  c  kˆ .
The vector projection may also be written (and interpreted) as the product of the vector magnitude and the
cosine of the angle between the vector and the corresponding coordinate axis. For instance (see the plot above),
cz  c  kˆ  c kˆ cos   c cos  .
The vector quantities cx ˆi, c y ˆj, cz kˆ , which may be interpreted as vector constituents of c , sum up, using the
rules of the vector addition, into the vector c itself:
c  cx ˆi  c y ˆj  cz kˆ .
The magnitude b  b of vector b in Cartesian coordinates is given by
b  b = bx 2  by 2  bz 2 .
 bx 
ˆ
ˆ
ˆ
Another common way to present Cartesian vector b  bx i  by j  bz k is to write it as  by  .
b 
 z
Vector c  b , negative of vector b  bx ˆi  by ˆj  bz kˆ , in the coordinate form will appear as
cx ˆi  c y ˆj  cz kˆ  c  bx ˆi  by ˆj  bz kˆ ,
or
 cx   bx 
 c    b  ,
 y  y
 c   b 
 z  z
so that c  b  0 .
7. Examples of operations with Cartesian vectors
Let b be a Cartesian vector and a be a scalar. Then, ab  abx ˆi  aby ˆj  abz kˆ .
Consider Cartesian vectors a  ax ˆi  a y ˆj  az kˆ , b  bx ˆi  by ˆj  bz kˆ , and their sum c  a  b  cx ˆi  c y ˆj  cz kˆ .
The sum of a and b in component form may be written as
a  b  (ax +bx )iˆ  (a y  by )jˆ  (az  bz )kˆ ,
so components of c  a  b are given by
cx  ax  bx , c y  a y  by , cz  az  bz .
Scalar (dot) product a  b of two Cartesian vectors a  ax ˆi  a y ˆj  az kˆ and b  bx ˆi  by ˆj  bz kˆ is given by
ˆ  (b ˆi  b ˆj  b k)
ˆ .
a  b  (ax ˆi  a y ˆj  az k)
x
y
z
Using the dot product properties (p. 5), we have
ˆ  a ˆj  (b ˆi  b ˆj  b k)
ˆ  a kˆ  (b ˆi  b ˆj  b k)
ˆ
a  b  ax ˆi  (bx ˆi  by ˆj  bz k)
y
x
y
z
z
x
y
z
 axbx ˆi  ˆi  axby ˆi  ˆj  axby ˆi  kˆ  a ybx ˆj  ˆi  a yby ˆj  ˆj  a ybz ˆj  kˆ  azbx kˆ  ˆi  azby kˆ  ˆj  azby kˆ  kˆ ,
which provides
a  b  axbx  a y by  azbz .
If a  b , the above expression produces already considered (in p. 5) formula for the vector magnitude. Indeed,
in this case:
b  b  b2  bx 2  by 2  bz 2 and b  b = bx 2  by 2  bz 2 .
Note that for perpendicular vectors a and b ( cos   0 ; see p. 5):
a  b  axbx  a y by  azbz  0 ,
and for parallel vectors a and b ( cos   1 ; see p. 5):
a  b  axbx  a y by  azbz  ab ,
where the plus sign corresponds to the vectors pointing in the same direction. Using a  a x 2  a y 2  az 2 and
b  bx 2  by 2  bz 2 , the condition of the vector parallelism becomes
ax a y az

 ,
bx by bz
(see also Class 6), with all ratios in the above expression being positive for vectors pointing in the same
direction and negative for vectors pointing in the opposite directions.
Streamwise Vorticity
Relative Helicity