Hi Hugo,


was

You are welcome!

> however, is how a cross product can be perpendicular to TWO vectors, and
how
> this can bring different results - in 2 dimensions.. If both vectors lie
> flat, because they are on a piece of paper, it's possible to put a third

I see your problem. The example I used for the cross product uses two
vectors within a flat plane as input. That's right. But the resulting vector
( which is the cross product) extents into the -third- dimension. Although
the input vectors lie flat on the floor, the result vector goes either up or
down. This is the only way to be perpendicular on both input vectors. No
other chance (except the input vectors are parallel or antiparallel [simply
spoken both lie along a straight line]).

> vector *everywhere* on the paper, being perpendicular to them.. hhmmm..

No you can't. Remember my very first line: vectors start always in the
origin. So all THREE vectors (input and result) share one common point: the
origin. Now you know where your vector has to start from. Try to find a
place for it's tip so that the whole thing is perpendicular to both vectors
(which form a "V" for example). The only way is up or down. Draw a big V
onto your sheet of paper (which symbolise two vectors starting at the
origin) and put your pencil onto the basepoint of the V. The pencil stays
with its lower end at this point. Move the tip of it to find a situation so
that the pencil is perpendicular to both vectors.
I think this makes it clear:
Since both vectors lie in one plane (your sheet of paper), the result vector
can only be perpendicular to both vectors at the same time if it is
perpendicular to exactly this PLANE. So the cross product will always be
perpendicular to your sheet. Regardless in which directions you rotate or
move it. The pencil rotates and moves as well and stays perpendicular to the
paper.

> Again, what is the visual ... purpose of cross products.

It's advantage is that you can define a plane by using two vectors and get
easily a normal vector onto it. This is very important to compute the angle
of an incoming light ray hitting our plane (as used for reflection) for
example. It depends on the problem you want so solve mathematically but
believe me there are a lot of situations where a vector which is standing
normal to a plane is very helpful. It's usage in computer graphics might be
to determine if you "see" the front side or the flip side of a flat piece of
the surface (e.g. on triangle of a mesh). This could be solved by using the
cross product since it direction relative to the plane is well defined and
you don't have to care about the individual input vectors.
Additionally you know the content of the area defined by the input vectors
by just measuring the cross product vector's length.

> head.. Maybe it's possible to make a simple illustration of a cross
product,
> not in 3 dimensions with 3 cylinders with each their color and direction,

I think a three dimensional model is the only way to get the cross product
right. I don't know if a cross product is even defined for two-dimensional
vectors. I don't think so. Some operations just need explicit prerequisites.
You can't explain addition or subtraction by just using one number....
An animation would be the best but I'm afraid I haven't time enough to make
a good movie. I have an idea but it takes it's time.
Anyway, if all failes I still can do that.

I'll try to describe it:
Imagine a big L. Take this L and rotate it's long axis. Watch the short
axis: it moves in a circle, creating a disk if rotated fast enough. The long
axis of this L would be a vector. The rotating short axis is the SUMMARY of
all possible vectors perpendicular to the it. (Remember all vectors start at
the origin therefore you can't move the connection point of the two axes).
Now imagine you replace the rotating axis by a disk. You get a thing similar
to a flat umbrella. This thing is our vector and a plane perpendicular to
it. ALL vectors perpendicular to our vector MUST lie within this plane,
within this two dimensions.
Create a second thing like this. This would be the second vector. Now you
got two umbrellas. Since all vectors start in the same point we have to
bring our vectors (together with their normal planes) together. Where should
they join? Exactly in the point where the vector hits the plane (since this
is the common point of the vector and all its normal vectors. Therefore it
must be the same point for the other vector!). Bring the umbrellas together
so that they share this point. What happens to the two planes? Correct, they
intersect. The intersection of two planes is a line. This line is the only
intersection of the normal vectors of BOTH input vectors. This intersection
is perpendicular to BOTH vectors since both normal planes share it. It is
our cross product!
Is this scenario more or less confusing? :o)

> to think the solution is obvious, "just do this!" but then, they can't
find
> a mathematical way of doing it, until  cross products suddenly are the
magic
> stick.

You can solve all these problems without using vectors as well. But doing it
without vectors results usually in a vast amount of sine, cosine, tanges and
their reverse relatives and hyperbolic cousins. Of course you would get the
same thing but the effort is very large, time consuming, easily misleading
and last but not least it takes tons of mathematical operations. Vectors are
just simpler and much nicer for that. All you need for (static) vectors is
basically what I described in my last post. As soon as you got used to the
dot and cross product nothing can stop you anymore ;-)

> class room.. :o)  Don't take me wrong please.. I enjoyed it, and I'll come
> back.

Great. As long as you have fun with it keep on asking!
Stupid questions don't exist. There are only stupid answers.

best regards

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