Graphics Reference
In-Depth Information
6.2.2 Multiplying a Vector by a Scalar
Given a vector
n
,2
n
means that the vector's components are doubled. For
example, if
⎡
⎤
⎡
⎤
3
4
5
6
8
10
⎣
⎦
⎣
⎦
n
=
then
2
n
=
which seems logical. Similarly, if we divide
n
by 2, its components are halved.
Note that the vector's direction remains unchanged - only its magnitude
changes.
It is meaningless to consider the addition of a scalar to a vector such as
n
+ 2, for it is not obvious which component of
n
is to be increased by 2. If all
the components of
n
have to be increased by 2, then we simply add another
vector whose components equal 2.
6.2.3 Vector Addition and Subtraction
Given vectors
r
and
s
,
r
±
s
is define as
⎡
⎤
⎡
⎤
⎡
⎤
x
r
y
r
z
r
x
s
y
s
z
s
x
r
±
x
s
⎣
⎦
⎣
⎦
r
⎣
⎦
r
=
s
=
±
s
=
y
r
±
y
s
(6.7)
z
r
±
z
s
Vector addition is commutative:
a
+
b
=
b
+
a
(6.8)
⎡
⎣
⎤
⎦
+
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
+
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
1
2
3
4
5
6
4
5
6
1
2
3
5
7
9
e
.
g
.
However, like scalar subtraction, vector subtraction is not commutative:
a
−
b
=
b
−
a
⎡
⎤
⎡
⎤
⎡
⎤
⎡
⎤
4
5
6
1
2
3
1
2
3
4
5
6
⎣
⎦
−
⎣
⎦
⎣
⎦
−
⎣
⎦
e
.
g
.
=
a
(6.9)
Let's illustrate vector addition and subtraction with two examples. Figure
6.5 shows the graphical interpretation of adding two vectors
r
and
s
. Note
that the tail of vector
s
is attached to the head of vector
r
. The resultant
vector
t
=
r
+
s
is defined by adding the corresponding components of
r
and
s
together. Figure 6.6 shows a graphical interpretation for
r
a
−
b
=
b
−
s
. This time the
components of vector
s
are reversed to produce an equal and opposite vector.
Then it is attached to
r
andaddedasdescribedabove.
−
