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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
ntuple
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 3space or the
xyz
space and denote an arbitrary vector in
R
3
by (
x
,
y
,
z
) instead of (
x
1
,
x
2
,
x
3
).
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View Full Document1.1.2
Remark
−
We usually write vectors in the form
x
= (
x
1
,
x
2
, .
..,
x
n
) as above. But
we could also write
x
as a column,
⎡
x
1
⎤
x
=
⎢
x
2
⎢
⎢
:
⎢
⎣
x
5
⎦
or as a diagonal if we wanted to.
•
The important thing is that the order of the components is respected.
•
If the vector is written horizontally we often specify this as say it is a "
row
vector
".
Operation
What we start with
:
What we get
:
Addition
2 vectors
x
and
y
in
R
n
:
x
= (
x
1
,
x
2
, .
..,
x
n
)
and
y
= (
y
1
,
y
2
, .
..,
y
n
).
1 vector
x + y
in
R
n
:
x + y
= (
x
1
+
y
1
,
x
2
+
y
2
, .
..,
x
n
+ y
n
)
Scalar multiplication
1 vector
x
in
R
n
:
x
= (
x
1
,
x
2
, .
..,
x
n
) and
one number
c
in
R
(called a
scalar
)
1 vector
c
x
in
R
n
:
c
x
= (
cx
1
,
cx
2
, .
..,
cx
n
).
Dot product
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This note was uploaded on 03/19/2009 for the course M m136 taught by Professor Fokshuen during the Spring '09 term at Waterloo.
 Spring '09
 FokShuen
 Algebra, Vectors, Scalar

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