Vector Products (this relates to an earlier question I had)

[Rockazella]

What is the real life significance of vector products? I understand how vector sums are used and related to real life situations, but not so much for products. The products are said to point perpendicular to the plane of the other 2 vectors. To me this sounds like more of a conceptual math idea that really can't be related to anything in the real world.

Yes, no?

 

[Alexander]

Magnetic field is one: it is just a vector product of observer's velocity and of electric field this observer is moving by: B=[vE]/c²

That is why magnetic force is such "twisted" force - because it is the result of existence of such "twisted" object as a vector product.

 

[Hurkyl]

Cross products seem abstract because they are are dual to the concept they actually describe.


Vectors are used to represent directions. They are good at representing a single direction, such as pointing along a line... however they are somewhat clumsy at representing a multidimensional direction, such as describing a plane.


The "natural" way to describe a planar direction is with something called a bivector. Just like a vector points along a direction and has a size, a bivector points along a planar direction and has a size. There is something called a "wedge product" that allows you to multiply two vectors to yield a bivector that contains them and has the appropriate size.


Now, it is somewhat more complicated to deal with bivectors, because they are a more complicated concept. However, in three dimensional space, we can simplify them via using a complementary space. Instead of using the bivector which describes a planar direction, we use the normal vector which describes the direction perpendicular to the planar direction. That is what the cross product computes.

 

[rdt2]

Originally posted by Hurkyl
Cross products seem abstract because they are are dual to the concept they actually describe.

All true but I get the impression that the original poster wasn't really at this stage, so at the risk of being simplistic:

The cross-product appears whenever there is rotation or a tendency for something to try to rotate.

Think of a whirlpool - the water can rotate clockwise or anti-clockwise about the centre. To allow us to perform calculations with both cases, we assign one a positive sign and the other a negative sign. If we're looking down on the pool from above, we take anticlockwise as positive and clockwise as negative. We can then describe the anticlockwise rotation by an arrow pointing upwards and whose length is proportional to the speed of rotation. Clockwise rotation is represented by a vector pointing downwards. So, the speed and direction of the rotation is represented by a vector normal to the surface.

Now consider a force F acting at some distance r from a central point; this produces a turning moment M about the centre, the magnitude of M is F.r.sin(w) (where w is the angle between F and r) and M is either clockwise or anticlockwise depending on the direction of F. Both of these cases are contained in the cross product:

M = r x F.

Beware: this only works in 3-D space, the only space in which there is one and only one axis (which may point up or down) at right angles to a plane.

 

[Alexander]

Originally posted by Hurkyl



Vectors are used to represent directions. They are good at representing a single direction, such as pointing along a line... however they are somewhat clumsy at representing a multidimensional direction, such as describing a plane.

Actually an element (portion) of surface or of plane IS a vector. Recall Gauss law (=scalar product of area with field).

 

[chroot]

Originally posted by Alexander
Actually an element (portion) of surface or of plane IS a vector. Recall Gauss law (=scalar product of area with field).

Actually, it is not -- which is the reason Hurkyl posted what he did. The only manifold in which you can get away with describing a plane with a vector is R^3. In general, you cannot use a vector to describe a plane; you'd use the wedge products of two vectors. In normal Euclidean 3-space R^3, the wedge product is the 'same' as a cross product. (Omitting the 'trivial' detail of chriality when demoting a 2-form to a 1-form.)

- Warren