MA 265 LECTURE NOTES: WEDNESDAY, MARCH 26
Inner Product Spaces
Direction in
R
2
and
R
3
.
We have seen that the angle between two vectors
u
and
v
in
R
2
is given by
θ
= arccos
u
1
v
1
+
u
2
v
2

u
 
v

where
0
≤
θ
≤
π.
Similarly, the angle between two vectors
u
and
v
in
R
2
is given by
θ
= arccos
u
1
v
1
+
u
2
v
2
+
u
3
v
3

u
 
v

where
0
≤
θ
≤
π.
We use this to make a few of definitions: We say that two nonzero vectors
u
and
v
are
•
in the
same direction
if
θ
= 0, i.e., 0
◦
;
•
orthogonal
or
perpendicular
if
θ
=
π/
2, i.e., 90
◦
; and
•
in
opposite directions
if
θ
=
π
, i.e., 180
◦
.
Inner Products on
R
2
and
R
3
.
There is a simple way to express the formula above. Define the
standard
inner product
or
dot product
as the scalar
u
·
v
=
(
u
1
v
1
+
u
2
v
2
on
R
2
;
u
1
v
1
+
u
2
v
2
+
u
3
v
3
on
R
3
.
Then we have the following formulas:
•
The
length
of a vector is

v

=
√
v
·
v
.
•
The
distance
from
v
to
u
is

v

u

=
p
(
v

u
)
·
(
v

u
) =
p

u

2

2 (
u
·
v
) +

v

2
•
The
angle
between
u
and
v
is
θ
= arccos
u
·
v

u
 
v

.
Hence it follows that two vectors
u
and
v
– not necessarily nonzero! – are
•
in the
same direction
if
u
·
v
=

u
 
v

;
•
orthogonal
or
perpendicular
if
u
·
v
= 0; and
•
in
opposite directions
if
u
·
v
=

u
 
v

.
Example.
Consider the vector
x
=

3
4
.
We compute those vectors which are (1) in the same direction as
x
, (2) orthogonal to
x
, and (3) in the
direction opposite to
x
. By looking at the geometry, we see that a vector in the same direction or opposite
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 Spring '10
 ...
 Scalar, Dot Product, inner product, Inner product space, Inner Products

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