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Cross product

  • Thread starter Haywire
  • Start date

Haywire

1. Homework Statement



2. Homework Equations



3. The Attempt at a Solution
I didn't use the template, because I am not having difficulties with a problem.

I am just starting to study rotational motion and there it appears the cross-product. I don't like to memorize formulae that I don't understand it's meaning.

Why is [tex]\vec a \times \vec b}[/tex] defined mathematically the way it is. Is there some trick to memorize?

Thanks in advance.
 

Hurkyl

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It's defined that way because it's useful.


If it helps, cross products are almost just like ordinary multiplication: if we write the standard basis vectors as i, j, and k, then all you need to know to compute a cross product is that

[tex]\begin{equation*}\begin{split}
i \times i = j \times j = k \times k = 0 \\
i \times j = k \\
j \times k = i \\
k \times i = j \\
j \times i = -k \\
k \times j = -i \\
i \times k = -j
\end{split}\end{equation*}[/tex]

(which is pretty easy to memorize), and that you can apply the distributive rule. (but not the associative rule, or the commutative rule!!!)

One geometric meaning to a cross products relates to perpendicularity -- you can already see that in the above identities. Another geometric meaning to the cross product of v and w is the "area" of the parallelogram with sides v and w, represented as a vector perpendicular to both v and w.
 
2,903
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Arg, I remember asking this question a while ago. It is NOT "defined that way because it's useful".....grrrrr. There is a GEOMETRICAL PROOF for why it works.

Here, I found it. Ta- DAAA:

http://www.math.oregonstate.edu/bridge/papers/dot+cross.pdf

Enjoy.
 
Last edited:

Haywire

Thank you both for your help. It is much more clear now for me. :)
 

Haywire

What can you tell me about this unit vector [itex]\vec e_\theta[/itex]? Sorry, for the double post.
 

cristo

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Haywire

Why is the direction of [tex]
\bold{\hat{\theta}} [/tex] that one?
 

cristo

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Why is the direction of [tex]
\bold{\hat{\theta}} [/tex] that one?
Since [itex]\theta[/itex] is the azimuthal angle, then [itex]\bold{\hat{\theta}}[/itex] is the unit vector in the azimuthal direction. You can think of it in the same way as, say, [itex]\bold{\hat{x}}[/itex] is the unit vector in the direction of the x axis, then [itex]\bold{\hat{\theta}}[/itex] is the unit vector in the direction of the azimuthal "axis." Since [itex]\theta[/itex] is the azimuthal angle, the unit vector is thus the tangent vector in the direction of the azimuthal angle.
 

Haywire

Thank you cristo! I can see it now.

Is your username some reference of The Count of Monte Cristo by Alexandre Dumas ?
 

cristo

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Science Advisor
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Thank you cristo! I can see it now.

Is your username some reference of The Count of Monte Cristo by Alexandre Dumas ?
You're welcome. Haha, no my username is my nickname, derived from my surname. I prefer your version though- sounds more sophisticated!
 
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What is all this talk of memorization? Just use the right hand rule and determinants. Cross-products involve no memorization.
 

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