84 Chapter^2
( c) We have noted that the line L: z = a + bt has the orientation of
vector b. Hence the line L': z = a' + b' T will be perpendicular to L iff b' =
i>.b, where >. =j:. 0 is some real number. This condition may be expressed as
and, alternatively, as
~ = i>.
bb' v
b + b =^0ororb'
Re-=0
bbb' + bb
1
= 0. (2.4-9)
(2.4-10)( d) In general, the oriented angle e from the line L to the line L', denoted
L(L, L'), is defined to be the oriented angle between the corresponding
vectors b and b'. Hence
.. b'
e = L(L,L') = argb' - argb = arg b (2.4-11)
3. Ray or oriented half-line with origin at a. This notion is defined by
the mapping h: z = a + bt, 0 S t < +oo. The graph of the ray is the
set of points
{z: z=a+bt,O:St<+oo}
- Oriented closed segment. The oriented closed segment, denoted
[z1, z2], from the point z 1 to the point z2 is defined by the mapping
z = (1 ·-t)z1 + tz2,
and its graph is the set
[z 1 , z 2 ]* = {z: z + (1 - t)z 1 + tz 2 , 0 :St :S 1}
(2.4-12)(2.4-13)which coincides with the straight-line segment joining z 1 to z 2 (Fig. 2.3).
In fact, if z is any point on the segment, we must have z - z 1 = t(z 2 - z 1 ),
where tis real and 0 St S 1. Hence it follows that z = (1 - t)z1 + tzz.
y(^0) x
Fig. 2.3