Vector
A vector is alternate notation for an
n tuple, so that
is the n-tuple
(
a
1
,
a
2
,
…
,
a
n
)
The length of a Vector
Given
x
∈
R
n
, then we define the it's length as
‖
x
‖
:=
∑
i
=
1
n
x
i
2
Dot Product
Given two sequences of numbers of the same length:
a
=
(
a
1
,
…
,
a
k
)
and
b
=
(
b
1
,
…
,
b
k
)
, their dot product denoted and defined by
a
⋅
b
=
∑
i
=
1
k
a
i
⋅
b
i
Note that
⋅
:
R
k
×
R
k
→
R
Dot Product Geometric
a
⋅
b
=
‖
a
‖
‖
b
‖
cos
(
θ
)
where
θ
is the angle between
a
,
b
Norm Squared is the Dot Product
For any
x
∈
R
n
we have that
‖
x
‖
2
=
x
⋅
x
Cross Product
Given two vectors
a
,
b
∈
R
3
we define their cross-product to be
a
×
b
:=
‖
a
‖
‖
b
‖
sin
(
θ
)
n
where
θ
is the angle between
a
and
b
and
n
is a unit vector such that
a
,
n
,
b
is positively oriented.
Collinear
Given a vector space
(
V
,
F
)
then we say that two vectors
x
,
y
∈
V
are collinear if there exists an
α
∈
F
such that
x
=
α
y
Parallelogram Law
Suppose
x
,
y
∈
R
n
then we have
‖
x
+
y
‖
2
+
‖
x
−
y
‖
2
=
2
‖
x
‖
2
+
2
‖
y
‖
2
‖
x
+
y
‖
2
+
‖
x
−
y
‖
2
=
(
x
+
y
)
⋅
(
x
+
y
)
+
(
x
−
y
)
⋅
(
x
−
y
)
=
x
⋅
x
+
2
x
⋅
y
+
y
⋅
y
+
x
⋅
x
−
2
x
⋅
y
+
y
⋅
y
=
2
x
⋅
x
+
2
y
⋅
y
=
2
‖
x
‖
2
+
2
‖
y
‖
2
The Norm of The Half of The Sum of Two Unit Vectors is One iff They are Equal
Suppose
x
,
y
∈
R
n
such that
‖
x
‖
=
‖
y
‖
=
1
then
‖
x
+
y
2
‖
=
1
⟺
x
=
y
⟹
Suppose that
‖
x
+
y
2
‖
=
1
we'd like to prove that
x
=
y
, thus it's equivalent to show that
‖
x
−
y
‖
=
0
, recall that from the
parallelogram law that
‖
x
−
y
‖
=
2
‖
x
‖
2
+
2
‖
y
‖
2
−
‖
x
+
y
‖
2
thus it's enough to show that
2
‖
x
‖
2
+
2
‖
y
‖
2
−
‖
x
+
y
‖
2
=
0
, which means we just have to show that
‖
x
+
y
‖
2
=
2
‖
x
‖
2
+
2
‖
y
‖
2
, moreover
x
,
y
were unit vectors so we have
‖
x
‖
=
‖
y
‖
=
1
and we now just need to prove that
‖
x
+
y
‖
2
=
4
. We've assumed that
‖
x
+
y
2
‖
=
1
, this means that
‖
x
+
y
‖
=
2
thus we conclude that
‖
x
+
y
‖
2
=
4
as needed.
⟸
Suppose that
x
=
y
, then
‖
x
+
y
2
‖
=
‖
2
x
2
‖
=
‖
x
‖
=
1
as needed.
Cosine Law
Suppose that
x
,
y
∈
R
n
and
θ
is the angle between them, then
‖
x
+
y
‖
2
=
‖
x
‖
2
+
2
‖
x
‖
‖
y
‖
cos
(
θ
)
+
‖
y
‖
2
Dot Product Is a Sum of Norms
x
⋅
y
=
‖
x
+
y
‖
2
−
‖
x
‖
2
−
‖
y
‖
2
2
−
><
!
−
−
2
3
]
+
[
3
2
1
]
, where
[
−
−
><
!
−
−
3
2
1
]
(the red dot) is a part of the line when
t
=
0
, and-->
−
><
!
−
−
]
-->