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

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The equation (2a + b)×(c - a) + (b + c)×(a × b) = a × c involves vector quantities and can be proven using properties of the cross product, such as associative, distributive, and anti-commutative properties. Participants suggest expanding the equation to simplify and cancel terms, noting that a × a equals zero due to the angle being zero. There is a discussion about the validity of the question, with some participants questioning its correctness based on dimensional analysis. A clarification is made regarding the use of sine instead of cosine in the context of the cross product. Overall, the conversation emphasizes the importance of careful manipulation of vector equations.
jiangxiaoyu
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All characters are vector quantities

The question is prove the following equation:

→ → → → → → → → → →
(2a + b)×(c -a ) + (b + c)×(a × b) = a × c
 
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Can you put forward any idea that you may have on solving this problem?
 
Why don't you use the asociative, distributive and anticonmutative properties of the cross product?
 
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?
 
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!
 
Last edited:
AiRAVATA and HallsofIvy already told you. Expand and it cancels out simply. Have you actually tried it?
 
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!
 
jiangxiaoyu said:
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.
 

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