\( \textbf{(1)} \left| \left| \vec{a} \times \vec{b} \right| \right|^2 = (a_2 b_3 - a_3 b_2 )^2 + (a_3 b_1 - a_1 b_3)^2 + (a_1 b_2 - a_2 b_1)^2 \)
(2)=a22b32−2a2a3b2b3+a32b22+a32b12−2a1a3b1b3+a12b32+a12b22−2a1a2b1b2+a22b12
(3)=(a12+a22+a32)(b12+b22+b32)−(a1b1+a2b2+a3b3)2
And so on.
How do we go from step (2) to step (3), algebraically?
#Algebra
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Comments
Brahmagupta-Fibonacci identity.
I fail to see why you'd torment yourself in such a way; an easier way would be to start with the Binet-Cauchy identity and the vector triple product...