Definition 13.2.
A number, i.e., an element of
Rfractur
, is referred to as a
scalar
.
So 1, 0,
π
,
e
and

5 are all examples of scalars.
We have defined two kinds of objects, vectors and scalars. The question
arises as to how we may combine these two things (e.g., can we add them
together?, multiply them together?, etc.). We now address this question.
Definition 13.3.
For any
vectoru
= (
u
1
, u
2
, ..., u
n
)
∈ Rfractur
n
and any
c
∈ Rfractur
(i.e.,
vectoru
is any vector and
c
is any scalar), the operation of
scalar multiplication of a
vector
is defined as:
cvectoru
= (
cu
1
, cu
2
, cu
3
, ..., cu
n

1
, cu
n
)
That is, to multiply the vector
vectoru
by the scalar
c
, we multiply each com
ponent of the vector by that scalar.
Example
1
.
For
vectoru
= (1
,
2
,
3),
vectorv
= (1
,

3
,
4
,
0) and
vectorw
= (1
,
1
,
2
,
2
,
3), find 3
vectoru
,

2
vectorv
and
π vectorw
.
Solution:
3
vectoru
=
3(1
,
2
,
3) = (3
,
6
,
9)

2
vectorv
=

2(1
,

3
,
4
,
0) = (

2
,
6
,

8
,
0)
and
π vectorw
=
π
(1
,
1
,
2
,
2
,
3) = (
π, π,
2
π,
2
π,
3
π
)
Definition 13.4.
The
zero vector
in
Rfractur
n
, denoted
vector
0, is the
n
vector whose
components are all zero.
For instance, in
Rfractur
2
, we have
vector
0 = (0
,
0), whereas in
Rfractur
3
,
vector
0 = (0
,
0
,
0) and
for
vector
0
∈ Rfractur
4
, we have
vector
0 = (0
,
0
,
0
,
0).
Definition 13.5.
The operation of
vector addition
is defined as follows:
For any
vectoru
= (
u
1
, u
2
, u
3
, ..., u
n

1
, u
n
) and
vectorv
= (
v
1
, v
2
, v
3
, ..., v
n

1
, v
n
), i.e., any
vectoru
,
vectorv
∈ Rfractur
n
, we define the
sum
of the two vectors to be:
vectoru
+
vectorv
= (
u
1
+
v
1
, u
2
+
v
2
, u
3
+
v
3
, ..., u
n

1
+
v
n

1
, u
n
+
v
n
)
That is, for vectors in the same space, we can perform the vector addition
vectoru
+
vectorv
by adding corresponding components of the vectors
vectoru
and
vectorv
.