Vector dot products : Check correctness?

Question

1) $\vec a.\vec b. \vec {(Ae)}=\vec b.\vec a. \vec {(Ae)}$

2) $(\vec a.\vec b) (\vec e.\vec f)=\vec a.(\vec b. \vec e). \vec f$

3) $\frac{e^TAf}{e^Tf}=\frac{\vec e.\vec {(Af)}}{\vec e.\vec f}=\frac{\vec a.\vec b}{\vec a.\vec b}\frac{\vec e.\vec {(Af)}}{\vec e.\vec f}$

4) $\frac{\vec a.\vec b}{\vec a.\vec b}\frac{\vec e.\vec {(Af)}}{\vec e.\vec f} \approx \frac{\vec e.\vec a}{\vec e.\vec a}\frac{\vec b.\vec {(Af)}}{\vec b.\vec f}$

4 seems wrong. But is there any condition where it might hold?

Answer 1

Dot product of 2 vectors is a scalar.

1) $\vec a.\vec b $ is a scalar and $\vec a. \vec b = \vec b . \vec a$ and $(\vec a.\vec b)\vec{Ae} = (\vec b.\vec a)\vec{Ae}$

2) 2nd equation doesn't always hold true.

For example, let $\vec a = \vec b = \hat i $ and $\vec e = \vec f = \hat j$

$\vec a . \vec b = 1$ and $\vec e . \vec f = 1$

$(\vec a . \vec b)(\vec e . \vec f) = 1$

But, $\vec b . \vec e = \hat i. \hat j = 0$