Question

if C = xA - yB where x=B.B and y=A.B, A,B,C are vectors. if C=0 and xA=yB, how do we conclude A or B is the scalar multiple of the other. Assume we have not defined what a dot product is and hence we do not know that dot product results in scalar. This is tom apostor's dot prod in section vector algebra. he proved cauchy's theorem with only the prodperties.

Answer 1

IF "we have not defined what a dot product is and hence we do not know that dot product results in scalar" then I struggle to see what xA actually means?
\[x \vec {A} = (\vec B • \vec B )\text{ } \vec A = \text{ ...}???\]

lets say [incorrectly] that B.B produces a vector Z, then this is meaningless too:

\[\vec Z \vec A = \vec Z \vec B\]
even if you take it a little further
\[|Z | \hat Z \vec A = |Z | \hat Z \vec B\]

unless \[\vec A = \vec B\]

or unless B.B is scalar

Answer 2

I get what you are saying. just xA does not make sense without operation defined.
I think i misunderstood the proof.
dot product operation on 2 vectors is defined by product of its components and hence a scalar. From this, we deduce properties of this dot product like association, commutaion etc.
so he is using x = B.B which by defnition is a scalar.
He is choosing C as xA - yB, difference of scalar multiples of A and B. As xA = yB, this can be true only if either one is a scalar multiple of the other. I think I am good now.
Thanks for your answer.

just curious, Why not just use scalar value instead of defining x=B.B and y = A.B. both are just the same.