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 )wheres=(−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 )⊥,wheret=−(
−−→
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 )⊥,wheret=− 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(