The problem involves proving the mixed product formula for vectors in R3, specifically showing that (A×B) . [(B×C)×(C×A)] equals (A,B,C)^2, where A, B, and C are vectors. The discussion centers around vector identities and properties of the cross and dot products.
The discussion is ongoing, with participants exploring various aspects of the problem. Some have identified terms that vanish, while others are questioning how these relate to the final expression. There is no explicit consensus yet, but productive lines of reasoning are being developed.
Participants are considering the implications of the properties of the dot and cross products, particularly in relation to the triple scalar product. There is a focus on ensuring all assumptions and definitions are clear as they work through the proof.
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?