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I am studying cross products and I just cannot get my head around the materials. Can someone explain to me why...

a) The cross product $\vec{a}\times \vec{b}$ is normal to $\vec{a}$ and $\vec{b}$. Where did this rule come from?

b) $|\vec{a}\times \vec{b}| = |\vec{a}||\vec{b}| \sin\theta$ is area of a parallelogram

c) $|\vec{a} \cdot (\vec{b} \times \vec{c})|$ is the volume of a parallelepiped.

These rules look very strange to me. I just don't see how someone could have come up with them out of thin air. There has to be some reasoning and/or insight I'm missing.

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Could you please post the definition of the cross product you have been given? –  AWertheim Jun 9 '13 at 20:11
    
The one you're calling "this rule" is actually a definition of cross product. As for the other ones, you can figure them out by trying to find those area and volume. –  Kaster Jun 9 '13 at 20:12
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Read through the wiki entry, perhaps it will get your confusion solved: en.wikipedia.org/wiki/Cross_product –  Shuhao Cao Jun 9 '13 at 20:15
    
@AWertheim : I wasn't given much of a definition from my physics book. The book says that : "The magnitude of $\vec{a} \times \vec{b}$ is the product of two vector magnitudes and the sine of the angle between them." and then it gives some properties of cross products –  Help Jun 9 '13 at 20:28

1 Answer 1

Items (a) and (b) in the question are, according to the book you quote in a comment, the definition of the cross-product; in fact, they should be supplemented by a right-hand rule, since they determine the product only up to a sign. So I see two essential parts to your question. One is how (c) enters the picture. The other is why (a), (b), and the right-hand rule are a reasonable definition and what is their motivation.

The question about (c) is relatively easy, because this formula follows from (a) and (b). Specifically, if $\vec a,\vec b,\vec c$ are three sides of a parallelepiped, then the volume of this parallelepiped is given by its base area times its height. If we set the parallelepiped down on the face spanned by $\vec b$ and $\vec c$, then the cross-product $\vec b\times\vec c$ of these has magnitude equal to the base area and is directed along the desired height. Meanwhile, the third side $\vec a$ might be at some strange angle, partly parallel and partly perpendicular to the base; its component perpendicular to the base, which gives the parallelepiped's height, is its length times the cosine of the angle between it and $\vec b\times\vec c$. Fortunately, the dot-product of two vectors is the product of their lengths times the cosine of the angle between them, so $\vec a\cdot(\vec b\times\vec c)$ is the volume of the parallelepiped. The absolute value in (c) is there because volumes are taken to be positive and I haven't been careful about the signs of vectors or orientations of parallelepipeds.

Now for the harder question: Why was this definition of cross-products invented in the first place, and why is it reasonable? There are various answers available, but I don't know the history. I'll just list some facts that might make the definition look better.

First, the cross-product has at least some nice algebraic properties. In particular, it is bilinear: $$ (\lambda\vec b)\times\vec c=\vec b\times(\lambda\vec c)=\lambda(\vec b\times\vec c) $$ and $$ (\vec b_1+\vec b_2)\times\vec c=(\vec b_1\times\vec c)+(\vec b_2\times\vec c) $$ and similarly when the sum is in the second factor. The first of these two formulas is pretty obvious, but the second is not; it would fail if, for example, we used some other trig function instead of the sine in (b).

There are deeper nice algebraic properties, summarizable by saying that the cross-product makes $\mathbb R^3$ into a Lie algebra.

That connects with geometry, in that this Lie algebra is the Lie algebra of the group $SO(3)$ of rotations of $\mathbb R^3$ about the origin.

Of course the parallelepiped formula (c) provides another connection with geometry.

The cross-product occurs naturally in several places in physics. For example, the force exerted by a magnetic field on a moving charge is proportional to the cross product of the field vector and the particle's velocity vector.

I'll stop here because I'm out of time, but there are other stories to be told, ranging from exterior algebra and quaternions in mathematics to rotational motion in physics. (I conjecture that the definition originally came from physics.)

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