Section 1.1
Solid Mechanics Part III
Kelly
3
1.1 Vector Algebra
1.1.1
Scalars
A physical quantity which is completely described by a single real number is called a
scalar
.
Physically, it is something which has a magnitude, and is completely described
by this magnitude.
Examples are
temperature,
density
and
mass
.
In the following,
lowercase (usually Greek) letters, e.g.
γ
β
α
,
,
, will be used to represent scalars.
1.1.2
Vectors
The concept of the
vector
is used to describe physical quantities which have both a
magnitude and a direction associated with them.
Examples are
force
,
velocity
,
displacement
and
acceleration
.
Geometrically, a vector is represented by an arrow; the arrow defines the direction of the
vector and the magnitude of the vector is represented by the length of the arrow, Fig.
1.1.1a.
Analytically, vectors will be represented by lowercase bold-face Latin letters, e.g.
a
,
r
,
q
.
The
magnitude
(or
length
) of a vector is denoted by
a
or
a
.
It is a scalar and must be
non-negative.
Any vector whose length is 1 is called a
unit vector
; unit vectors will
usually be denoted by
e
.
Figure 1.1.1: (a) a vector; (b) addition of vectors
1.1.3
Vector Algebra
The operations of addition, subtraction and multiplication familiar in the algebra of
numbers (or scalars) can be extended to an algebra of vectors.
a
b
c
(a)
(b)

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Section 1.1
Solid Mechanics Part III
Kelly
4
The following definitions and properties fundamentally
define
the vector:
1.
Sum of Vectors:
The addition of vectors
a
and
b
is a vector
c
formed by placing the initial point of
b
on the terminal point of
a
and then joining the initial point of
a
to the terminal
point of
b
.
The sum is written
b
a
c
+
=
.
This definition is called the
parallelogram law for vector addition because, in a geometrical interpretation of
vector addition,
c
is the diagonal of a parallelogram formed by the two vectors
a
and
b
, Fig. 1.1.1b.
The following properties hold for vector addition:
a
+
b
=
b
+
a
… commutative law
a
+(
b
+
c
)
=
(
a
+
b
)
+
c
… associative law
2.
The Negative Vector:
For each vector
a
there exists a
negative vector
.
This vector has direction
opposite to that of vector
a
but has the same magnitude; it is denoted by
a
−
.
A
geometrical interpretation of the negative vector is shown in Fig. 1.1.2a.
3.
Subtraction of Vectors and the Zero Vector:
The
subtraction
of two vectors
a
and
b
is defined by
)
(
b
a
b
a
−
+
=
−
, Fig.

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