Geometry and Trigonometry 607And this is thecab-bacidentity once we notice that−→a ·
−→
b =−^12 tr(AB).
581.An easy computation shows that the mapf:R^3 →su( 2 ),f (x, y, z)=(
−iz y−ix
y+ix iz)
,
has the desired property.582.Denoting by−→
A,
−→
B,
−→
C,
−→
A′,
−→
B′,
−→
C′the position vectors of the vertices of the two
triangles, the condition that the triangles have the same centroid reads
−→
A+−→
B+
−→
C=
−→
A′+
−→
B′+
−→
C′.
Subtracting the left-hand side, we obtain
−−→
AA′+−−→
BB′+
−−→
CC′=
−→
0.
This shows that−−→
AA′,
−−→
BB′,
−−→
CC′form a triangle, as desired.583.Set−→v 1 =−→
AB,−→v 2 =−→
BC,−→v 3 =−→
CD,−→v 4 =−→
DA,−→u 1 =−−→
A′B′,−→u 2 =−−→
B′C′,
−→u
3 =−−→
C′D′,−→u 4 =−−→
D′A′. By examining Figure 72 we can write the system of equations2 −→v 2 −−→v 1 =−→u 1 ,
2 −→v 3 −−→v 2 =−→u 2 ,
2 −→v 4 −−→v 3 =−→u 3 ,
2 −→v 1 −−→v 4 =−→u 4 ,in which the right-hand side is known. Solving, we obtain−→v 1 =^1
15−→u 1 +^2
15−→u 2 +^4
15−→u 3 +^8
15−→u 4 ,Figure 72