Let
a
=
h
a
1
,a
2
,a
3
i
and
b
=
h
b
1
,b
2
,b
3
i
be vectors and
θ
the angle between them.
Length:

a

=
p
a
2
1
+
a
2
2
+
a
2
3
.
Dot product: a
·
b
=
a
1
b
1
+
a
2
b
2
+
a
3
b
3
=

a

b

cos
θ
.
a
is parallel to
b
if
a
=
c
b
, for some scalar
c
.
a
and
b
are perpendicular if
a
·
b
= 0.
Vector projection
of
b
onto
a
:
proj
a
(
b
) = comp
a
(
b
)
u
, where
u
=
a
/

a

is a
unit vector and comp
a
(
b
) =
b
·
u
is the component of
b
in the direction of
a
.
Cross product: a
×
b
=
h
a
2
b
3

a
3
b
2
, a
3
b
1

a
1
b
3
, a
1
b
2

a
2
b
1
i
, if
±
±
±
±
a
b
c
d
±
±
±
±
=
ad

bc
a
×
b
=
±
±
±
±
±
±
i
j
k
a
1
a
2
a
3
b
1
b
2
b
3
±
±
±
±
±
±
=
±
±
±
±
a
2
a
3
b
2
b
3
±
±
±
±
i

±
±
±
±
a
1
a
3
b
1
b
3
±
±
±
±
j
+
±
±
±
±
a
1
a
2
b
1
b
2
±
±
±
±
k
,

a
×
b

=

a

b

sin
θ
,
a
×
b
is perpendicular to both
a
and
b
.
The area of the triangle with
a
and
b
as two edges is
A
=

a
×
b

/
2.
The volume of the parallelepiped with
a
,
b
and
c
as three edges is
V
=

a
·
(
b
×
c
)

.
a
×
b
= 0 if
a
and
b
are parallel.
a
·
(
b
×
c
) = 0 if
a
,
b
and
c
lie in the same plane.
Parametric equations of a line:
x
=
x
0
+
at
,
y
=
y
0
+
bt
,
z
=
z
0
+
ct
where
(
x
0
,y
0
,z
0
) is on the line and
h
a,b,c
i
is parallel to it:
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 Fall '07
 BoonYeap
 Derivative, ∂x ∂z ∂y, Chain Rule Case

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