Chapter 3
Vectors
In physics we have parameters that can be completely described by a
number and are known as
scalars
.
Temperature and mass are such
parameters.
Other physical parameters require additional information about direction and
are known as
vectors
.
Examples of vectors are displacement, velocity, and
acceleration.
In this chapter we learn the basic mathematical language to describe
vectors.
In particular we will learn the following:
Geometric vector addition and subtraction
Resolving a vector into its components
The notion of a unit vector
Addition and subtraction vectors by components
Multiplication of a vector by a scalar
The scalar (dot) product of two vectors
The vector (cross) product of two vectors
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•
Scalar
 one dimensional quantity, that is, quantities
which require only one number to giving its size or
magnitude
.
•
Examples of scalar quantities are:
–
Temperature
Time
Speed
Mass
Location Along a Line (1D)
•
Vector
 multidimensional quantity. Multi
dimensional quantities are those which require more
than one number to completely describe them.
Vectors, unlike scalars, have two characteristics,
magnitude and direction
.
•
Examples of vector quantities are:
–
Location in a Plane (2D)
Location in Space (3D)
Velocity
Acceleration
Force
An example of a vector is the displacement vector,
which describes the change in position of an object as it
moves from point
A
to point
B
. This is represented by an
arrow that points from point
A
to point
B
.
The length of
the arrow is proportional to the displacement magnitude.
The direction of the arrow indicated the displacement
direction.
The three arrows from
A
to
B
, from
A'
to
B'
, and from
A''
to
B''
, have the same magnitude and direction. A vector
can be shifted without changing its value if its length and
direction are not changed.
In books vectors are written in two ways:
Method 1:
(using an arrow above)
Method 2:
a
(using boldface print)
The magnitude of the vector is indicated by italic print:
a
.
a
r
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Geometric Vector Addition
Sketch vector
using an appropriate scale.
Sketch vector
using the same scale.
Place the tail of
at the tip of .
The vector
starts from the tail of
and terminates at the tip of
a
b
b
a
s
a
s
a
b
=
+
r
r
r
r
r
r
r
r
r
commutative:
Negative
of a gi
.
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 Spring '08
 FALCO
 Vectors, Vector Space, Dot Product, Mass, ax

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