Why is the cross product defined the way it is?

[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 $$\bold{e_\theta} \equiv \bold{\hat{\theta}}$$.

 

[Haywire]

Why is the direction of $$\bold{\hat{\theta}}$$ that one?

 

[cristo]

Why is the direction of $$\bold{\hat{\theta}}$$ 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.