FACT SHEET FOR 20E EXAM 1
1.
From Chapter 1
•
Dot product of two vectors
v
= (
v
1
, v
2
, . . . , v
n
) and
w
= (
w
1
, w
2
, . . . , w
n
):
v
·
w
=
v
1
w
1
+
v
2
w
2
+
. . .
+
v
n
w
n
.
•
Cross produce of two vectors in 3D (its a vector!):
v
×
w
= (
v
2
w
3

v
3
w
2
, v
3
w
1

v
1
w
3
, v
1
w
2

v
2
w
1
)
.
We have that
v
·
(
v
×
w
) =
w
·
(
v
×
w
) = 0 and the length
v
×
w
is the
area of the parallelogram spanned by
v
and
w
.
The direction obeys the
“right hand rule”.
•
The equation for a line through the vector
x
0
and in the direction
v
0
is
(
t
) =
x
0
+
t v
0
.
•
The matrix product of an
n
×
m
matrix (
a
ij
) and an
m
×
k
matrix (
b
ij
) is
the
n
×
k
matrix (
c
ij
) with
c
ij
=
∑
m
l
=1
a
il
b
lj
.
•
The determinant of a 2
×
2 matrix (
a
ij
) is

A

=
a
11
a
22

a
12
a
21
. This is the
area of the parallelogram spanned by the vectors (
a
11
, a
12
) and (
a
21
, a
22
).
2.
From Chapter 2
•
The derivative matrix of the vectorvalued function
Φ
:
R
n
→
R
k
is the
matrix
D
Φ

x
0
= (
∂
j
f
i
(
x
0
)), where the
f
i
are the component functions of
Φ
.
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 Spring '09
 STAFF
 Derivative, Vectors, Dot Product, right hand rule

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