Monday January 5
−
Lecture 1 : Review :
Vectors
in
R
2
and
R
3
.
(Refers to section
1.1 and 1.2 in your text)
Objectives
:
1. Define vectors in
R
n
and in particular in
R
2
and
R
3
.
2. Define scalars, coordinates, components.
3. Represent vectors in
R
2
and
R
3
as directed line segments.
4.
Add, subtract, scalar multiply, dot product
R
n
.
5.
Add, subtract, scalar multiply vectors when represented by directed line segments.
6.
Recognize equivalent vectors, collinear, parallel vectors.
7.
Define a linear combination of vectors, unit vector, midpoint of two points or
vectors in
R
2
and
R
3
.
8.
Give the standard basis of vectors for
R
3
.
1.1
Introduction
−
The elements,
x
= (
x
1
,
x
2
), of the Cartesian plane
R
2
are familiar and
usually referred to as
ordered pairs
, or
coordinates
of the plane.
•
In this course we will begin referring to these as members of a larger family of
mathematical objects called
vectors
.
•
Observe how we represent vectors by a letter in
boldface
, while scalars are
simply expressed in
italics
.
1.1.1
Definition
−
Let
R
n
=
{(
x
1
,
x
2
, ...,
x
n
)
:
x
1
,
x
2
, ...,
x
n
∈
R
}. We say that
x
is an
n-tuple
of real numbers and we call
x
a
vector
.
The
x
i
's are called the
entries
,
coordinates
or
components
of the vector. To distinguish vectors from scalars, we
represent vectors in boldface.
•
If we write
x
is a vector in
R
5
, we mean
x
= (
x
1
,
x
2
,
x
3
,
x
4
,
x
5
) where
x
1
,
x
2
,
x
3
,
x
4
and
x
5
are real numbers.
•
It is also common practice, whenever it useful to do so, to refer to a vector in
R
2
and
R
3
as a
point
. Hence to say, "let (
−
1, 2, 0) be a
point
in
R
3
" means the same
thing as saying "let (
−
1, 2, 0) be a
vector
in
R
3
".
•
Occasionally we may refer to
R
2
as the
xy
-plane and denote an arbitrary vector in
R
2
by (
x
,
y
). Whenever we find it useful we might refer to
R
3
as 3-space or the
xyz
-space and denote an arbitrary vector in
R
3
by (
x
,
y
,
z
) instead of (
x
1
,
x
2
,
x
3
).