Hi Hugo,

It is difficult to explain these kind of things without using drawings.
About the "shadow of the vector", it is not very important for you to
understand it right now, but let me see if I can give a clearer explanation:
If you have a 3d vector, (imagine an arrow in 3d space) and you have a light
from the top of room, it will cast a shadow of the arrow in the floor. That
shadow, is the projection of the vector in the floor plane. The dot product
allows you to find this projection.

About your doubt concerning the "visualization" of the dot product, I
remember I had the same feeling in my first course in linear algebra. We are
used to the natural interpretation of the usual product between to numbers,
but the dot product fails to have the same natural interpretation. If you
see the second formula for the dot product, |A|*|B|*cos(angle), you can see
that the dot product can be very useful because it contains the angle
between the two vectors. The first formula lets you calculate the dot
product between A and B very easily. Then, |A| and |B|, the norms of the
vectors (the lengths) can be calculated very easily too. Finally, you can
isolate the cosine of the angle, which can be very useful.

Maybe you can visualize the dot product as a "weighted" measure of the angle
between two vectors.

About your other question, in which you cannot see how can the cross product
can be perpendicular to the other two vectors, YOU'RE RIGHT! That is because
the cross product is defined only for 3d vectors.
You cannot have 3 vectors perpendicular to each other in 2D space, but you
can have them in 3D space.

I hope this helps you,

Fernando.

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