John Ellison, UCR
p.1
Notes on Chapter 1 (pp. 1126)
Vectors
To describe motion in three dimensions we will use
vectors
. A
vector
has magnitude and direction. An example of a vector quantity
is displacement. The displacement of a particle which moves from
point A to point B can be represented by a vector. Its magnitude is
the length of the line joining A and B, and its direction is from A to
B, indicated by an arrow. Other examples of vector quantities are
velocity and acceleration.
A
scalar
is a quantity which has magnitude only (no direction).
Examples of scalar quantities are speed, temperature, mass, time, and
pressure.
We will represent vectors by italic symbols with an arrow above, e.g.
. The magnitude of a vector is represented by the symbol with no
arrow, e.g.
a
, or by the notation
. Now consider two
displacements, one from A to B, and the other from B to C. The net
displacement is from A to C. This can be represented graphically as
shown in the figure at top right. We call AC the
vector sum
(or
resultant
) of the vectors AB and AC. We can represent this by the
vector addition equation
which says that vector
is the vector sum of vectors
and
. Note
that this is not a simple algebraic addition  it involves the
magnitudes and directions of the vectors, not just the magnitudes.
Vector addition has two important properties:
Vector subtraction
is defined by:
where the vector
is a vector with the same magnitude as
but
with the opposite direction (see the figure below).
s
=
a
b
a
b
=
b
a
(commutatative law)
a
b
c
=
a
b
c
(associative law)
a
d
a
−
b
a
−
b
s
a
b
∣
a
∣
−
b
b
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentJohn Ellison, UCR
p.2
Unit Vectors
The
component
of a vector is
the projection of the vector on an axis
(see figure below). The
x component
and
y component
are the
projections on the
x
axis and
y
axis respectively, and are denoted by
a
x
and
a
y
, respectively. The process of finding the components of a
vector is called
resolving the vector
. From simple geometry (see the
figure below), we find that the components are given by:
where
θ
is the angle the vector
makes with the positive direction
of the
x
axis.
In order to express a vector in terms of its components we define
unit vectors
as vectors of unit magnitude which point in the
direction of the positive
x
,
y
,
z
directions, respectively. We will
normally use a coordinate system in which the +
x
direction is
horizontal, the +
y
direction is vertical, and the +
z
direction is "out of
the page". Such a coordinate system (shown in the figure above left)
is said to be a
righthanded coordinate system
. Now we can
express the vectors
as
This says that vector
is equal to the sum of a vector of magnitude
a
x
along the +
x
direction and a vector of magnitude
a
y
along the +
y
direction (and similarly for
). The magnitude and angle of vector
are
a
=
a
x
i
a
y
j
b
=
b
x
i
b
y
j
a
a
and
b
b
a
x
=
a
cos
and
a
y
=
a
sin
magnitude:
a
=
a
x
2
a
y
2
This is the end of the preview.
Sign up
to
access the rest of the document.
 Winter '08
 Ellison
 Vectors, Dot Product, John Ellison

Click to edit the document details