Graphics Programs Reference
In-Depth Information
A
Vector Products
It is trivial to add and subtract vectors, but vectors can also be multiplied. This short
appendix is a reminder of (or a refresher on) the two important operations of dot product
and cross product.
The dot product (or inner product) of two vectors is denoted by
P
•
Q
and is defined
as the scalar
(
P
x
,P
y
,P
z
)(
Q
x
,Q
y
,Q
z
)
T
=
PQ
T
=
P
x
Q
x
+
P
y
Q
y
+
P
z
Q
z
.
This simple definition implies that the dot product is commutative,
P
•
Q
=
Q
•
P
,
and is also distributive with respect to vector addition or subtraction,
P
•
(
Q
±
T
)=
P
•
Q
±
P
•
T
.
The dot product also has a simple and useful geometric interpretation; it equals
|
P
||
Q
|
cos
θ
,where
θ
is the angle between the vectors. The dot product of perpendicular
(or
orthogonal
) vectors is therefore zero. We use Figure A.1 to prove this interpretation.
Part a shows a triangle with three sides
a
,
b
,and
c
and three angles
A
,
B
,and
C
opposite
those sides. We draw a line from vertex
B
that is perpendicular to side
b
. This line
divides the triangle into two right-angle triangles. The three sides of the triangle on the
right are
a
,
a
sin
C
,and
a
cos
C
, while the sides of the triangle on the left are
c
,
a
sin
C
,
and
b
−
a
cos
C
. Applying Pythagoras's theorem to the latter triangle yields the law of
cosines
c
2
=(
a
sin
C
)
2
+(
b
a
cos
C
)
2
−
=
a
2
sin
2
C
+
b
2
2
ab
cos
C
+
a
2
cos
2
C
=
a
2
(sin
2
C
+cos
2
C
)+
b
2
−
−
2
ab
cos
C
=
a
2
+
b
2
2
ab
cos
C.
This extends the Pythagorean theorem to arbitrary triangles.
Given two arbitrary vectors
a
and
b
separated by an angle
θ
, Figure A.1b shows
how we can subtract them to obtain a third vector
c
=
a
−
−
b
, such that
a
,
b
and
c
form
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