R2 13 + 12 = R1-1 (p 11) (5.7)
and taking the inner-product with vector z provides
(z R2 13) + (z 12) = (z R1-1 p) (z RI-1 11). Applying (4.5) to the first term of both sides of this equation gives (after rearranging terms) R1 p R2-1 z 13 = z RI- 11 + z 12. (5.8) The right hand side of Eq. (5.8) is constant since
ai
Ri-1 i = dici and z.1i=di are independent of ei.
diri
Multiplying Eq. (5.7) by R2-1 gives
13 + R2-1 12 = R2-1 RI-1 (p 11) (5.9)
and multiplication of Eq. (5.5) by R2-1 R1-1 yields
R3 z = R2-1 R1-1 t. (5.10)
The inner product of corresponding sides of equations (5.9) and (5.10) produces
(13 + R2-1 12).(R3 z) = [R2- R1-1 (p 11)].[R2-1 RI-1 t]. Repeated use of properties (4.4) and (4.5) and reordering simplifies this last equation to
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