The discussion centers on proving the mixed product formula for vectors in R3, specifically that (A×B) . [(B×C)×(C×A)] equals (A, B, C)², where A, B, and C are vectors in R3. The key equation used is W×(U×V)=(W . V) U - (W × U) V. Participants clarify that terms like (B×C) . C and (A×B) . A vanish, leading to the conclusion that [(B×C) . A] and [(A×B) . C] are equal, thus confirming the relationship to the triple scalar product.
PREREQUISITESMathematics students, physics students, and anyone studying vector calculus or linear algebra who seeks to deepen their understanding of vector identities and their applications in R3.
JasonHathaway said:Homework Statement
Prove that (A×B) . [(B×C)×(C×A)]=(A,B,C)^2
where A, B, C are vectors in R3.
Homework Equations
W×(U×V)=(W . V) U - (W × U) V
The Attempt at a Solution
Assuming K=(A×B), M=(B×C):
K . [M×(C×A)]
K . [(M . A) C - (M . C) A]
[(M . A)(K . C) - (M . C)(K . A)]
Then:
[(B×C) . A] [(A×B) . C] - [(B×C) . C] [(A×B) . A]
JasonHathaway said:(b×c) . C = (c×b) . C = (c×c) . B = (0) . B = 0
JasonHathaway said:(A×B) . A will vanish as well, and I'll end up with [(B×C) . A] and [(A×B) . C] which are equal.
But how - by algebra - can they be equal to (A,B,C)^2?