Understanding the Meaning and Importance of Cross Product in Rotational Motion

[Haywire]

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

Thanks in advance.

 

[Hurkyl]

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

\begin{equation*}\begin{split}<br /> i \times i = j \times j = k \times k = 0 \\<br /> i \times j = k \\<br /> j \times k = i \\<br /> k \times i = j \\<br /> j \times i = -k \\<br /> k \times j = -i \\<br /> i \times k = -j<br /> \end{split}\end{equation*}

(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.

 

[Cyrus]

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.

 

[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 \vec e_\theta? Sorry, for the double post.

 

[cristo]

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

Check out : http://mathworld.wolfram.com/SphericalCoordinates.html There's a good pic here, and the equations for the unit vectors. Note that <br /> \bold{e_\theta} \equiv \bold{\hat{\theta}}.

 

[Haywire]

Why is the direction of <br /> \bold{\hat{\theta}} that one?

 

[cristo]

Why is the direction of <br /> \bold{\hat{\theta}} that one?

Since \theta is the azimuthal angle, then \bold{\hat{\theta}} is the unit vector in the azimuthal direction. You can think of it in the same way as, say, \bold{\hat{x}} is the unit vector in the direction of the x axis, then \bold{\hat{\theta}} is the unit vector in the direction of the azimuthal "axis." Since \theta 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]

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!

 

[Mindscrape]

What is all this talk of memorization? Just use the right hand rule and determinants. Cross-products involve no memorization.