Hi,


the one looking for cross and dot products.. Fernando gave a short
description.. I liked how you start by pointing out the VECTOR input and
VECTOR / NUMBER output of each function.. A good start.. You also tell about
the usefullness of the functions, which increase our attention.

Regarding the dot product, I had to skip the sentence "the shadow of a
vector in a plane".. It's still nonsense to me, but alright..You go on to
formulas, with the most simple being at first: A dot B = a1*b1 + a2*b2 +
a3*b3  Thank you!  This leaves no confusion in my mind, except that I can't
visualize how the result relates to the original vectors A and B.. I just
see multiplication and addition going on (things I know of).. You proceed
with calculating a dot product by using other formulas, involving radians,
theta, sqrt and exponentation.. This mostly goes beyond me, I still can't
see the "result".. You end the lesson with a good sentence, saying "if A and
B are orthogonal (perpendicular), then cos(theta)=0 and then the dot product
is equal to zero".. Very good.. This brings back a feeling of sense, order
and logic in my mind about the whole thing.. Order, because it seems there's
a formula behind this, that I WILL be able to undestand someday.. All in
all, I wasn't left in the land of strangers, thank you.  :o)


for your long explanation! I like that you began with simple things like
translation.. It's good to begin with something most people already knows
and understand, because it kind of makes us relax.. We're travelling with
you in the same train, whatever ... on the same level.. Then you came to
explaining dot and cross products:

I admit the landscape changed from familiar to unknown territory.. So it's
necessary for me to slow down, to follow.. Fortuntately I have your lesson
on text so I can read it as many times as I like.. :o)  I like how you
sticked to 2 dimensions on a piece of paper.. You tell that cross products
face either upwards or downwards, depending if the calculation is done
clockwise or counter-clockwise (something all the clever people talk about
when they compute smooth_triangles).. What I don't really understand,
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
vector *everywhere* on the paper, being perpendicular to them.. hhmmm..
Again, what is the visual ... purpose of cross products.

Anyway you used plain english, and I really appriciate your efforts!
Obviously you have done efforts, and english is not your native language (or
mine).. I will read your description again and try to get a picture in my
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,
it's nonsense..  :o)  I still haven't found a place on the net, where a good
illustration is found.. For example, a problem on paper that leads everyone
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.

Annnnd, thanks for reading my long explanation of how I felt, being in your
class room.. :o)  Don't take me wrong please.. I enjoyed it, and I'll come
back.

Regards,
Hugo

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