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1
CHAPTER 6
ORTHOGONALITY
Example:
Perpendicularity in
R
2
: the Pythagorean Theorem
algebraic condition for perpendicularity in
R
2
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Properties:
1.
0
is orthogonal to every vector in
R
n
.
2. For
u
,
v
∈
R
3
,
u
and
v
are perpendicular if and only if
u
•
v
= 0.
3. The dot product can also be represented by the matrix product
4. For
u
∈
R
n
,
v
∈
R
m
, and
A
∈
R
m
×
n
,
Proof
All results may be proven easily using the definition.
Corollaries:
1. For
c
∈
R
,
u
,
v
∈
R
n
,
c
u
•
v
has a clear meaning: (
c
u
)
•
v
or
c
(
u
•
v
).
2. For
c
1
,
c
2
,
…
,
c
p
∈
R
,
u
,
v
1
,
v
2
,
…
,
v
p
∈
R
n
,
3
Example:
Proof
= 0 if and only if
u
and
v
are orthogonal
Corollary:
Proof
= 0
⇔

u

=

v

Orthogonal projection of a vector onto a line
v
: any vector in
R
2
u
: any nonzero vector on
L
w
: orthogonal projection of
v
onto
L
,
w
=
c
u
z
:
v
−
w
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⇒
and
Distance from
P
(tip of
v
) to
L
:
Example:
L
is
y
= (1/2)
x
in
R
2
and
v
= [ 4
1 ]
T
.
Let
u
= [ 2
1 ]
T
.
Then
and the distance from the tip of
v
to
L
is
*
Proof
The inequality holds when
u
=
0
.
Assume
u
≠
0
.
Then
.
0
)
(
)
(
:
Note
.
4
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
=
−
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
≥
−
+
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This note was uploaded on 10/16/2010 for the course EE 155 taught by Professor Fong during the Spring '09 term at National Taiwan University.
 Spring '09
 Fong

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