Chapter 1
Vector Algebra
1.1 Terminology and Notation
Scalars
are mathematics quantities that can be fully defined by specifying their mag
nitude in suitable units of measure. Mass is a scalar quantity and can be expressed
in kilograms, time is a scalar and can be expressed in seconds, and temperature is a
scalar quantity that can be expressed in degrees Celsius.
Vectors
are quantities that require the specification of magnitude, orientation,
and sense. The characteristics of a vector are the magnitude, the orientation, and the
sense.
The
magnitude
of a vector is specified by a positive number and a unit having
appropriate dimensions. No unit is stated if the dimensions are those of a pure num
ber.
The
orientation
of a vector is specified by the relationship between the vector
and given reference lines and/or planes.
The
sense
of a vector is specified by the order of two points on a line parallel to
the vector.
Orientation and sense together determine the
direction
of a vector.
The
line of action
of a vector is a hypothetical infinite straight line collinear with
the vector.
Displacement, velocity, and force are examples of vectors quantities.
To distinguish vectors from scalars it is customary to denote vectors by boldface
letters Thus, the displacement vector from point
A
to point
B
could be denoted as
r
or
r
AB
. The symbol

r

=
r
represents the magnitude (or module, norm, or absolute
value) of the vector
r
. In handwritten work a distinguishing mark is used for vec
tors, such as an arrow over the symbol,
→
r
or
→
AB
, a line over the symbol, ¯
r
, or an
underline,
r
.
The vectors are most frequently depicted by straight arrows. A vector represented
by a straight arrow has the direction indicated by the arrow. The displacement vector
from point
A
to point
B
is depicted in Fig. 1.1(a) as a straight arrow. In some cases
it is necessary to depict a vector whose direction is perpendicular to the surface
1
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1 Vector Algebra
in which the representation will be drawn. Under this circumstance the use of a
portion of a circle with a direction arrow is useful. The orientation of the vector is
perpendicular to the plane containing the circle and the sense of the vector is the
same as the direction in which a righthanded screw moves when the axis of the
screw is normal to the plane in which the arrow is drawn and the screw is rotated
as indicated by the arrow. Figure 1.1(b) uses this representation to depict a vector
directed out of the reading surface toward the reader.
v
r
A
(a)
(b)
B
Fig. 1.1
Representations of vectors
A
bound
vector is a vector associated with a particular point
P
in space (Fig. 1.2).
The point
P
is the
point of application
of the vector, and the line passing through
P
and parallel to the vector is the line of action of the vector. The point of appli
cation may be represented as the tail, Fig. 1.2(a), or the head of the vector arrow,
Fig. 1.2(b). A
free
vector is not associated with any particular point in space. A
transmissible
(or
sliding
) vector is a vector that can be moved along its line of ac
tion without change of meaning.
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 Spring '08
 Clark,B
 Statics, Linear Algebra, Vectors, Vector Space, Dot Product

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