Sec. 11.6 Mobile coordinates 207
For anyZ∈Π,Z=O,wedefine−→
OZ
⊥
=−−→
OWwhereZ≡(x,y),W≡(−y,x),and
call this theGrassmann supplementof
−→
OZ. This clearly has the properties(−−→
OZ 1 +
−−→
OZ 2 )⊥=
−−→
OZ 1
⊥
+−−→
OZ 2
⊥
,
(k−OZ→)⊥=k(−OZ→)⊥,(−→
OZ
⊥
)⊥=−−→
OZ.
11.6.2 .....................................
InFwe take|O,I|=|O,J|=1. If|O,Z|=1andθis the angle inAFwith support
|IOZ, then we recall from 9.2.2 thatZ≡(x,y)wherex=cosθ,y=sinθ.AsI≡
( 1 , 0 ),J≡( 0 , 1 ), we note that
−→
OI⊥
=−→
OJ,
−→
OZ=cosθ−→
OI+sinθ−→
OJ,
−→
OZ
⊥
=−sinθ−→
OI+cosθ−→
OJ.
Suppose that we also have−OW−→=cosφ−OI→+sinφ−OJ→. Then by 11.4.1 we have
−−→
OW=r
−→
OZ+s−→
OZ
⊥
where
r=cosφcosθ+sinφsinθ,s=sinφcosθ−cosφsinθ.By the addition formulae we recognise that
r=cosFZOW,s=sinFZOW.11.6.3 Handling a triangle ..........................
Although for a triangle[Z 1 ,Z 2 ,Z 3 ], we have the vector form for the centroid as
(11.5.3) and for the incentre as (11.5.5) where as usuala=|Z 2 ,Z 3 |,b=|Z 3 ,Z 1 |,
c=|Z 1 ,Z 2 |, neither this formula for the incentre, nor the more awkward formula for
the orthocentre, are convenient for applications and generalization. In 7.2.3 we noted
that a unique circle passes through the vertices of a triangle[Z 1 ,Z 2 ,Z 3 ]. It is called the
circumcircleof this triangle and its centre is called thecircumcentre. It is possible
to find an expression for the circumcentre in terms of the vertices as in 11.4.4 but it
is tedious to cover all the cases. For these reasons we consider the following way of
representing any triangle.
O
W Z
90 ◦
Figure 11.12. Grassmann supplement.
Z (^1)
Z 2
Z 3
W 1
w 1 −z 2 =p(z 3 −z 2 );
z 1 −w 1 =qı(z 3 −z 2 )