Proving Cross Product distributivity, but

In summary, the cross product is only defined as: A X B = |A| |B| sin \theta times a unit vector perpendicular to the plane of A&B (direction according to the right hand rule, in the usual way).
  • #1
erogard
62
0
... with the cross product being only defined as: A X B = |A| |B| sin [tex]\theta[/tex] times a unit vector perpendicular to the plane of A&B (direction according to the right hand rule, in the usual way).
where theta is the smallest angle between vectors A & B.

A X ( B + C ) = A X B + A X C
is the equation I have to prove without using a component-wise approach; I'm considering the case where one of the three vectors would be perpendicular to the two other as I have shown the coplanar case.

I have tried using several approaches, graphical and algebraic, both unsuccessful so far.

If you have any ideas please let me know, just to get started don't do anything more than suggesting something - I should be able to figure it out.

Thanks!
 
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  • #3
Sadly I am not supposed to use anything that would deal with the operations applied to their components since I'm not supposed to have defined the cross product in such a way yet.
 
  • #4
erogard said:
Sadly I am not supposed to use anything that would deal with the operations applied to their components since I'm not supposed to have defined the cross product in such a way yet.


Well, with the definition you are given, you essentially have to prove

[tex]|\textbf{B}+\textbf{C}|\sin \theta_{A,B+C} = |\textbf{B}|\sin\theta_{A,B}+|\textbf{C}|\sin\theta_{A,C}[/tex]

Do you see why?
 
  • #5
gabbagabbahey said:
Well, with the definition you are given, you essentially have to prove

[tex]|\textbf{B}+\textbf{C}|\sin \theta_{A,B+C} = |\textbf{B}|\sin\theta_{A,B}+|\textbf{C}|\sin\theta_{A,C}[/tex]

Do you see why?

but doesn't it contradict the triangular inequality? This (on my little drawing, at least) would be equivalent to showing that two sides of a triangle add up to the length of the third.

I'll pause here and rethink about it for a minute.
 
  • #6
If you took away the sin(theta)s then it would be problematic, but those are all going to be different numbers
 

1. What is the cross product?

The cross product is a mathematical operation between two vectors that results in a vector perpendicular to both of the original vectors. It is denoted by the symbol "×" and is also known as the vector product.

2. What does it mean to prove cross product distributivity?

Proving cross product distributivity means showing that the cross product of two vectors multiplied by a third vector is equal to the cross product of each vector multiplied by the third vector individually. This property is an important property of vector operations and is used in many applications in physics and engineering.

3. How can cross product distributivity be proven?

Cross product distributivity can be proven using the properties of vector operations and the properties of the cross product. This involves breaking down the equation into smaller parts and using algebraic manipulations to show that both sides of the equation are equal.

4. Why is cross product distributivity important?

Cross product distributivity is important because it allows us to simplify complex vector equations and make calculations easier. It also has many applications in fields such as mechanics, electromagnetism, and computer graphics.

5. Are there any limitations to cross product distributivity?

Yes, there are some limitations to cross product distributivity. It only applies to vectors in three-dimensional space and cannot be extended to higher dimensions. It also does not hold for non-commutative vector operations, such as the triple product.

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