Page 105 - A Course in Linear Algebra with Applications

4.1: Examples of Vector Spaces 89

space. To achieve this, choose an arbitrary point I with co-
ordinates (u\, «2> U3) as the initial point of the line segment.
The end point of the segment is the point E with coordinates
(ui+ai, U2 + a 2, U3 + 0,3). The direction of the line segment
IE is indicated by an arrow:

a
a
w
E(u 1 + a 1 ,u 2 + 2. 3 + 3)

\{U :,U 2,U 3)

The length of IE equals

I — J a\ + 02 + 03

and its direction is specified by the direction cosines

cii/l, a 2/l, a 3/l.

Here the significant feature is that none of these quantities
depends on the initial point I. Thus A is represented by in-
finitely many line segments all of which have the same length
and the same direction. So all the line segments which repre-
sent A are parallel and have equal length. However the zero
vector is represented by a line segment of length 0 and it is
not assigned a direction.
Having connected elements of R 3 with line segments, let
us see what the rule of addition in R 3 implies about line seg-
ments. Consider two vectors in R 3

A = a 2 and B