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

  • Thread starter jiangxiaoyu
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In summary, the conversation is about proving the equation (2a + b)×(c -a ) + (b + c)×(a × b) = a × c. Suggestions are given to use the associative, distributive, and anticommutative properties of the cross product to simplify the equation. An example is given to explain the anticommutative property and it is pointed out that the question may be wrong due to a possible mistake in the equation. The conversation ends with the suggestion to expand and cancel terms to solve the problem.
  • #1
jiangxiaoyu
8
0
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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  • #2
Can you put forward any idea that you may have on solving this problem?
 
  • #3
Why don't you use the asociative, distributive and anticonmutative properties of the cross product?
 
  • #4
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?
 
  • #5
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:
  • #6
AiRAVATA and HallsofIvy already told you. Expand and it cancels out simply. Have you actually tried it?
 
  • #7
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!
 
  • #8
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.
 

1. What is the purpose of proving this equation?

The purpose of proving this equation is to demonstrate that it is mathematically valid and can be used as a reliable tool for solving problems in various fields of science and mathematics.

2. What are the steps involved in proving this equation?

The steps involved in proving this equation may vary depending on the specific approach and mathematical concepts used. However, some common steps may include simplifying both sides of the equation, applying mathematical operations and principles, and demonstrating that both sides are equal to each other.

3. How can this equation be applied in real-world scenarios?

This equation can be applied in various real-world scenarios, such as calculating the area of a rectangle, solving problems related to forces and motion, and determining the solutions to equations involving variables.

4. What is the significance of the variables (a, b, c) in this equation?

The variables (a, b, c) represent unknown quantities that can take on any numerical value. They allow for the equation to be generalized and used in a wide range of mathematical problems.

5. Are there any limitations to using this equation in mathematical calculations?

Every mathematical equation has its limitations, including this one. It may not be applicable in certain scenarios or may require additional assumptions or conditions for it to be valid. It is important to understand the context and limitations of the equation before using it in calculations.

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