# Cross Product Properties Question

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1. Sep 1, 2016

### Sho Kano

1. The problem statement, all variables and given/known data
$A\cdot B\times C\quad =\quad 2\\ (2A+B)\quad \cdot \quad [(A-C)\quad \times \quad (2B+C)]\quad =\quad ?$

2. Relevant equations
Various cross product and dot product properties

3. The attempt at a solution
I've only managed to get so far, don't really know what to do next
$A\cdot B\times C\quad =\quad 2\\ (2A+B)\quad \cdot \quad [(A-C)\quad \times \quad (2B+C)]\\ \\ =(2A+B)\quad \cdot \quad [(A-C)\times 2B\quad +\quad (A-C)\times C]\\ =(2A+B)\quad \cdot \quad [A\times 2B\quad -\quad C\times 2B\quad +\quad A\times C]$

2. Sep 1, 2016

### haruspex

The scalar triple product has a very useful cyclic property. X.(YxZ)=Y.(ZxX)=Z.(XxY). Switching the cyclic order swaps the sign.

3. Sep 2, 2016

### Sho Kano

I tried that already, the problem is I end up getting this mess after distributing
$2A\cdot A\times 2B\quad -\quad 2A\cdot C\times 2B\quad +\quad 2A\cdot A\times C\quad +\quad B\cdot A\times 2B\quad -\quad B\cdot C\times 2B\quad +\quad B\cdot A\times C$
and there's no way of using the triple scalar product to simplify that, other than the second term

4. Sep 2, 2016

### haruspex

The first, third, fourth and fifth terms simplify so much using that hint that they disappear immediately. Post an attempt at using it on the first term.

5. Sep 2, 2016

### Sho Kano

OH they simplify to 0. I was too focused on matching the given information with the terms.
$2A\cdot A\times B\\ =\quad 2[A\cdot A\times B]\\ =\quad 2[A\times A\cdot B]\\ =\quad 0$
So we are left with
$-2A\cdot C\times 2B\quad +\quad B\cdot A\times C\\ =\quad -2A\cdot 2[C\times B]\quad +\quad B\cdot A\times C\\ =\quad -4[A\cdot C\times B]\quad +\quad B\cdot A\times C\\ =\quad -4[A\times C\cdot B]\quad +\quad A\times C\cdot B\\ =\quad -3[A\times C\cdot B]\\ =\quad -3[A\cdot C\times B]\\ =\quad -3[-(A\cdot B\times C)]\\ =\quad -3(-2)\\ =\quad 6$

6. Sep 2, 2016

Looks right.

7. Sep 2, 2016

### Sho Kano

Thanks, it turned out to be so simple!