±
²
³
13
MATH
FACTS
101
13
13.1
Vectors
13.1.1
Deﬁnition
We
use
the
overhead
arrow
to
denote
a
column
vector,
i.e.,
a
linear
segment
with
a
direction
.
For
example,
in
threespace,
we
write
a
vector
in
terms
of
its
components
with
respect
to
a
reference
system
as
⎧
⎫
⎪
2
⎪
⎨
⎬
±a
=
1
.
⎪
⎪
⎩
⎭
7
The
elements
of
a
vector
have
a
graphical
interpretation,
which
is
particularly
easy
to
see
in
two
or
three
dimensions.
1.
Vector
addition:
+
±
b
=
±
c
⎧
⎫
⎧
⎫
⎧
⎫
⎪
2
⎪
⎪
3
⎪
⎪
5
⎪
⎨
⎬
⎨
⎬
⎨
⎬
1
+
3
=
4
.
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
⎩
⎭
⎩
⎭
7
2
9
Graphically,
addition
is
stringing
the
vectors
together
head
to
tail.
2.
Scalar
multiplication:
⎧
⎫
⎧
⎫
⎪
2
⎪
⎪
−
4
⎪
⎨
⎬
⎨
⎬
−
2
×
1
=
−
2
.
⎪
⎪
⎪
⎪
⎩
7
⎭
⎩
−
14
⎭
13.1.2
Vector
Magnitude
The
total
length
of
a
vector
of
dimension
m
,
its
Euclidean
norm,
is
given
by
²
m

±x

=
´
x
2
i
.
i
=1
This
scalar
is
commonly
used
to
normalize
a
vector
to
length
one.
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²
²
²
²
13
MATH
FACTS
102
13.1.3
Vector
Dot
or
Inner
Product
The
dot
product
of
two
vectors
is
a
scalar
equal
to
the
sum
of
the
products
of
the
corre
sponding
components:
m
±x
·
±
y
=
T
±
y
=
x
i
y
i
.
i
=1
The
dot
product
also
satisﬁes
·
±
y
=


±
y

cos
θ,
where
θ
is
the
angle
between
the
vectors.
13.1.4
Vector
Cross
Product
The
cross
product
of
two
threedimensional
vectors
and
±
y
is
another
vector
±
z
,
×
±
y
=
±
z
,
whose
1.
direction
is
normal
to
the
plane
formed
by
the
other
two
vectors,
2.
direction
is
given
by
the
righthand
rule,
rotating
from
to
±
y
,
3.
magnitude
is
the
area
of
the
parallelogram
formed
by
the
two
vectors
–
the
cross
product
of
two
parallel
vectors
is
zero
–
and
4.
(signed)
magnitude
is
equal
to

±y

sin
θ
,
where
θ
is
the
angle
between
the
two
x

±
vectors,
measured
from
to
±
y
.
In
terms
of
their
components,
⎧
⎫
²
i
j
ˆ
²
⎪
i
⎪
ˆ
ˆ
k
⎪
(
x
2
y
3
−
x
3
y
2
)
ˆ
⎪
²
²
⎨
⎬
×
±
y
=
²
x
1
x
2
x
3
²
=
(
x
3
y
1
−
x
1
y
3
)
ˆ
j
.
²
²
⎪
⎪
⎪
⎪
²
y
1
y
2
y
3
²
⎩
(
x
1
y
2
−
x
2
y
1
)
k
ˆ
⎭
13.2
Matrices
13.2.1
Deﬁnition
A
matrix,
or
array,
is
equivalent
to
a
set
of
column
vectors
of
the
same
dimension,
arranged
side
by
side,
say
⎡
⎤
23
⎢
⎥
A
=[
±a
±
b
]=
⎣
13
⎦
.
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 Spring '05
 GeorgeKocur
 Linear Algebra, Dot Product, Det

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