Graphics Reference
In-Depth Information
u
u
+
v
u
+
v
v
v
v
u
u
(a)
(b)
Figure 3.2
(a) The result of adding two vectors
u
and
v
is obtained geometrically by placing
the vectors tail to head and forming the vector from the tail of the first vector to the head
of the second. (b) Alternatively, by the parallelogram law, the vector sum can be seen as the
diagonal of the parallelogram formed by the two vectors.
3.3.1
Vector Arithmetic
The
sum
w
of two vectors
u
and
v
,
w
=
u
+
v
, is formed by pairwise adding the
components of
u
and
v
:
w
=
u
+
v
=
(
u
1
,
u
2
,
...
,
u
n
)
+
(
v
1
,
v
2
,
...
,
v
n
)
=
(
u
1
+
v
1
,
u
2
+
v
2
,
...
,
u
n
+
v
n
).
Geometrically, the vector sum can be seen as placing the arrow for
v
at the tip of
u
and defining their sum as the arrow pointing from the start of
u
to the tip of
v
.
This geometric view of the vector sum is often referred to as the
parallelogram law
of vector addition
, as the vector forming the sum corresponds to the diagonal of the
parallelogram formed by the two given vectors, as illustrated in Figure 3.2.
The subtraction of vectors,
w
=
−
u
v
, is defined in terms of the addition of
u
and the negation of
v
; that is,
w
v
of a vector
v
is a
vector of equal magnitude but of opposite direction. It is obtained by negating each
component of the vector:
=
u
+
(
−
v
). The
negation
−
−
v
=−
(
v
1
,
v
2
,
...
,
v
n
)
=
(
−
v
1
,
−
v
2
,
...
,
−
v
n
).
Componentwise, the subtraction of two vectors is therefore given by
w
=
u
−
v
=
(
u
1
,
u
2
,
...
,
u
n
)
−
(
v
1
,
v
2
,
...
,
v
n
)
=
(
u
1
−
v
1
,
u
2
−
v
2
,
...
,
u
n
−
v
n
).
Vectors can also be scaled through the multiplication of the vector by a constant
(Figure 3.3). The resulting vector
w
,
w
=
k
v
,froma
scalar multiplication
by
k
is
given by
w
=
k
v
=
k
(
v
1
,
v
2
,
...
,
v
n
)
=
(
kv
1
,
kv
2
,
...
,
kv
n
).
