Mathematics of Physics and Engineering

Euclidean Space as a Linear Space 5

Let r be the position vector for a point P. Consider another frame with
origin O'. Let r' be the position vector of P relative to O'. Now, let v be
the position vector of O' relative to O. The three vectors form a triangle
OO'P; see Figure 1.1.2. This suggests that we write r = v + r'. To get
from O to P we can first go from O to O' along v and then from O' to
P along r'. This can be depicted entirely with position vectors at O if
we move r' parallel to itself and place its initial point at O. Then r is a
diagonal of the parallelogram having sides v and r', all emanating from O.
This is called the parallelogram law for vector addition. It is a geometric
definition of v + r'. Note that the same result is obtained by forming the
triangle OO'P.

r'l S^T IT' r = v + r'

O v a

Fig. 1.1.2 Vector Addition

Now, consider three position vectors, u,v,w. It is easy to see that the
above definition of vector addition obeys the following algebraic laws:

u + v = v + u (commutativity)

(it + v) + w = u + (v + w) (associativity; see Figure 1.1.3) (1.1.1)

it + 0 = 0 + tt = u

The zero vector 0 is the only vector with zero length and no specific direc-
tion.
Next consider two real numbers, A and JJL. In vector algebra, real num-
bers are called scalars. The vector Ait is the vector obtained from it by
multiplying its length by |A|. If A > 0, then the vectors it and Ait have
the same direction; if A < 0, then the vectors have opposite directions. For
example, 2it points in the same direction as it but has twice its length,
whereas —u has the same length as u and points in the opposite direction
(Figure 1.1.4).