Graphics Reference
In-Depth Information
v
q
v
v
u
⋅
v
u
⋅
u
q
=
v
- -----
u
q
q
u
u
u
⋅
v
u
⋅
u
p
u
⋅
v
u
p
= -----
u
d
= ----- =
v
cos
q
(a)
(b)
Figure 3.5
(a) The distance of
v
along
u
and (b) the decomposition of
v
into a vector
p
parallel and a vector
q
perpendicular to
u
.
·
=
θ
u
v
u
v
cos
2
u
·
u
=
u
u
·
v
=
v
·
u
u
·
(
v
±
w
)
=
u
·
v
±
u
·
w
r
u
·
s
v
=
rs
(
u
·
v
)
3.3.5
The Cross Product
The
cross product
(or
vector product
) of two 3D vectors
u
=
(
u
1
,
u
2
,
u
3
) and
v
=
(
v
1
,
v
2
,
v
3
) is denoted by
u
×
v
and is defined in terms of vector components as
u
×
v
=
(
u
2
v
3
−
u
3
v
2
,
−
(
u
1
v
3
−
u
3
v
1
),
u
1
v
2
−
u
2
v
1
).
The result is a vector perpendicular to
u
and
v
. Its magnitude is equal to the product
of the lengths of
u
and
v
and the sine of the smallest angle
θ
between them. That is,
u
×
v
=
n
u
v
sin
θ
,
where
n
is a unit vector perpendicular to the plane of
u
and
v
. When forming the
cross product
w
v
, there is a choice of two possible directions for
w
. Here, and
by convention, the direction of
w
is chosen so that it, together with
u
and
v
, forms
a
right-handed coordinate system
. The
right-hand rule
is a mnemonic for remembering
what a right-handed coordinate system looks like. It says that if the right hand is
=
u
×
