Mathematical Physics Vol 1

Chapter 1. Vector algebra

28

It can be concluded that the two collinear (or parallel) vectors are linearly dependent, since α and β are different from zero. Thus, it can be said that all vectors k a , for arbitrary and real k and a̸ = 0, form a one-dimensional (1–D) real linear vector space. Such terminology is used due to the fact that to each point on the axis a position vector 13 can be assigned and conversely, to each vector from this set a point on the axis corresponds (one-to-one correspondence).

Consider now two non-collinear vectors a and b . Let us represent them by oriented segments with a com mon beginning O (Fig. 1.16). An arbitrary vector c , lying in the plane of vectors a and b , can be represented in the form c = m a + n b . (1.30) This relation follows from the vector addition rules and from the definition of multiplication of a vector by a scalar. From relation (1.30), similar as in the case of (1.28) and (1.29), and assuming:

c = m a + n b

n b

b

a

m a

Figure 1.16: Non-collinear vectors.

α γ

β γ

m = −

n = −

(1.31)

,

,

we obtain

α a + β b + γ c = 0 , (1.32) which is the condition for linear dependence of a set of three vectors, because not all constants are zero. In this way, each point in the plane can be determined by a position vector c , i.e. by a combination of the vectors m a + n b , where a and b are two linearly independent vectors, and m and n are the corresponding real numbers. Therefore, it can be said that the combination m a + n b defines a two-dimensional (2–D) real linear vector space. It can also be noted that in a 2–D linear vector space a set of three vectors is always linearly dependent. Consider now three non-coplanar 14 vectors a , b i c , starting from a common origin O (sl. 1.17).

d = m a + n b + p c

n b

b

p c

a

c

m a

Figure 1.17: Sum of vectors in 3 − D .

As in the previous cases, any subsequent vector d can be represented by the relation

d = m a + n b + p c ,

(1.33)

A = −→ OA , that starts in the origin O and ends in point A .

13 The position vector of a point A is the vector r 14 Vectors are coplanar if they are all parallel to one plane.