1. ## vectors

I want to proof that, given vector a non zero and a vector b, so
$\displaystyle a.b=0 <=> \exists c: c\times a=b$

I know how to proof <=

but how to proof =>?
My try:
$\displaystyle a.b=0 => \exists c: c\times a=b$
$\displaystyle <=> \sim \exists c: c\times a=b => \sim a.b=0$
$\displaystyle <=> \forall c: c\times a \neq b => a.b\neq 0$

$\displaystyle Supose \forall c: c\times a \neq b$
I tried to prove $\displaystyle a.b \neq 0$

Now $\displaystyle Supose \forall c: c\times a = b$
so b is orthogonal to both a and c
so $\displaystyle a.b=0$

So if $\displaystyle \forall c: c\times a \neq b$ we have $\displaystyle a.b \neq 0$ my doubt is, my I affirm this???

2. ## Re: vectors

Originally Posted by Pipita
I want to proof that, given vector a non zero and a vector b, so
$\displaystyle a.b=0 <=> \exists c: c\times a=b$
I know how to proof <=
but how to proof =>?
Assume that $a\cdot b=0$ and $a\ne 0$. Now define $c=\dfrac{-(b\times a)}{||a||^2}$

Does that work?

4. ## Re: vectors

Plato post #2 is correct. But define, out of the clear blue sky?

axb is perpendicular to a and b and (axb)xa is then in direction b.

(axb)xc=(a.c)b - (a.b)c from vector algebra, look it up.
(axb)xa=(a.a)b - (a.b)c, which gives Plato's formula, c=axb/a2