236 Vector and complex-number methods Ch. 11
Exercises
11.1 Show that ifZ 1 =Z 2 , the pointsZon the perpendicular bisector of[Z 1 ,Z 2 ]are
those for which
−→
OZ=^12 (
−−→
OZ 1 +
−−→
OZ 2 )+t(
−−→
OZ 2 −
−−→
OZ 1 )⊥,forsomet∈R.
11.2 Ifl=Z 1 Z 2 ,thenZ 4 =πl(Z 3 )if and only ifZ 4 =Z 1 +s(Z 2 −Z 1 )where
s=
(−OZ−→ 3 −−OZ−→ 1 ).(−OZ−→ 2 −−OZ−→ 1 )
‖
−−→
OZ 2 −
−−→
OZ 1 ‖^2
,
and also if and only if
−−→
OZ 4 =
−−→
OZ 3 +t(
−−→
OZ 2 −
−−→
OZ 1 )⊥,where
t=−
(
−−→
OZ 3 −
−−→
OZ 1 ).(
−−→
OZ 2 −
−−→
OZ 1 )⊥
‖
−−→
OZ 2 −
−−→
OZ 1 ‖^2
.
11.3 Ifl=Z 1 Z 2 ,thenZ 4 =sl(Z 3 )if and only if
−−→
OZ 4 =
−−→
OZ 3 +t(
−−→
OZ 2 −
−−→
OZ 1 )⊥,where
t=− 2
(
−−→
OZ 3 −
−−→
OZ 1 ).(
−−→
OZ 2 −
−−→
OZ 1 )⊥
‖
−−→
OZ 2 −
−−→
OZ 1 ‖^2
.
11.4 IfZ 2 ∈[Z 0 ,Z 1 andlis the mid-line of the angle support|Z 1 Z 0 Z 2 ,thenZ∈lif
and only if
−→
OZ=
−−→
OZ 0
+t
[
1
‖
−−→
OZ 2 −
−−→
OZ 0 ‖
(
−−→
OZ 2 −
−−→
OZ 0 )−
1
‖
−−→
OZ 1 −
−−→
OZ 0 ‖
(
−−→
OZ 1 −
−−→
OZ 0 )
]⊥
,
for somet∈R.
11.5 Prove that
−→
OZ
′
−
−−→
OZ 1 =cosα(
−→
OZ−
−−→
OZ 1 )+sinα(
−→
OZ−
−−→
OZ 1 )⊥, represents a
rotation about the pointZ 1 , through the angleα.
11.6 For any vector−OZ→, withZ≡F(x,y),define
−→
OZ
=
−→
OU,
−→
OZ
=
−→
OV,
whereU≡F(x,−y),V≡F(y,x),sothatU=sOI(Z),V=sl(Z),lbeing the
mid-line of|IOJ. Prove that
(i)
(
−−→
OZ 1 +
−−→
OZ 2 )=
−−→
OZ 1
+
−−→
OZ 2
,
(ii)
(k
−→
OZ)=k(