If a, b, c and d are not equal or 0 and (a.b)c=(b.c)a`,
show that a and b are parallel.
Since the dot product is a scalar, I can see that c and a are scalar multiples of each other, but what if b is prependicular to a and c? Wont the given equation still be satisfied whilst allowing a and b to be anything in the plane perpendicular to b? What am I doing wrong here?
Sorry for the poor formatting. I have no idea how it works
What you are saying is correct. In $\mathbb R^{3}$ let $e_1,e_2,e_3$ be the standard basis. Take $a=e_1,b=e_2$ and $c=e_3$. Then the equation holds but $a$ and $c$ are linearly independent.