Proving Equation (2a+b)×(c-a)+(b+c)×(a×b)=a×c

[jiangxiaoyu]

All characters are vector quantities

The question is prove the following equation:

→ → → → → → → → → →
(2a + b)×(c -a ) + (b + c)×(a × b) = a × c

 

[neutrino]

Can you put forward any idea that you may have on solving this problem?

 

[AiRAVATA]

Why don't you use the asociative, distributive and anticonmutative properties of the cross product?

 

[HallsofIvy]

It looks very simple to me by just doing what AiRAVATA says. As you would expect, a loot of things cancel. Look especially carefully at the "anti-commutative" property since you may not be used to that. In particular, what is axa?

 

[jiangxiaoyu]

a x a <=> |a|*|a|*cosθ ，
because θ=0 so |a|*|a|*cosθ =0 , a x a =0
Can you give a sample example relative my question? I can not find any example on my schoolbook.
Thanks you!

 

[KTC]

AiRAVATA and HallsofIvy already told you. Expand and it cancels out simply. Have you actually tried it?

 

[jiangxiaoyu]

Hi
The reasult is the question is wrong. Someone was sure the question was wrong at frist glance. The question can not be allowd by dimension role.

(b + c)×(a × b) may change to (b + c)×(a + b)

In that case, the question was very easy.
Thanks!

 

[Moo Of Doom]

a x a <=> |a|*|a|*cosθ ，
because θ=0 so |a|*|a|*cosθ =0 , a x a =0

Surely you mean sinθ, not cosθ. Just wanted to alert you of this mistake... also, that equation only measures the length, since the cross product yields a vector.