I need some help with this vectors question?

[dawdw]

So far I only got this, I have no clue where to go from here. The question and where I'm at is posted below. Please help me to get the answer.

 

[pka]

I object to the way you are writing the notation.
Either use caps or little letters for vectors, not both..
The notation \(\bf a\cdot b\times c\) is meaningless.
Surely if must be \(\bf a\cdot (b\times c)\).
Please correct and repost.

 

[dawdw]

So far I only got this, I have no clue where to go from here. The question and where I'm at is posted below. Please help me to get the answer.

 

[pka]

In the attachment you state: \(\bf(a\times b)\times (b\times c)=b\).
How do you know that?
We know that \(\bf u\times (v\times w)=(u\cdot w)v-(u\cdot v)w\)
\(\bf(a\times b)\times (b\times c)=[(a\times b)\cdot c]b-[(a\times b)\cdot b]c\)
Don't we know that \(\bf (a\times b)\bot b~?\) What does that tell us about \(\bf (a\times b)\cdot b~?\)