Question

If $a\times b=b\times c\ne 0$ with $a\ne -c$ then show that $a+c=kb$, where $k$ is scalar.

Answer 1

$a\times b=b\times c$

$a\times b=-c\times b$

$a\times b+c\times b=0$

$(a+c)\times b=0$

So, there are three posibilities:

(i)a+c=0

$\Rightarrow$a=-c which is not the case

(ii)b=0

In this case, $a\times b=0$ too, which is not the case

(iii)a+c= kb ,where k is scalar

which is the only possible scenario.

Hence,a+c= kb ,where k is scalar