Understanding Vector Multiplication: Why Does Cross Product Result in a Vector?

[f24u7]

Homework Statement

This is actually a concept question, but since its kind of elementary i post it here

I understand the calculation of the cross product, what i do not understand is why the cross product that only involve in 2 dimension will have the result of 3rd dimension

Homework Equations

A cross B = AB sin

A dot B = AB cos

The Attempt at a Solution

If i analyze the equation, i find that Asin is equal to A's y component, and if you times that with B, it will only result in a vector perpendicular to B and has the magnitude of A's y component times B, how does that end up with vector that is perpendicular to both A and B

and the right hand rule doesn't explain it either, it just shows how to obtain the direction of the third vector

besides, how does a difference in trigonometric function made cross product a vector, and scalar product a scalar?

from my understanding, AB cos is just like AB sin, it only gives a number, so where does the direction comes from

the concept of vector multiplication is really confusing, i hope someone can help explaining this

Thanks in advance

 

[CompuChip]

You need to be a little more careful in distinguishing vectors from magnitudes. The precise equations are

A x B = |A| |B| sin(x) (1)
where x is the angle between the vectors A and B, and |A| and |B| are just the magnitude (without direction information) of A and B.
Similarly,
A · B = |A| |B| cos(x) (2)

The difference is, that A · B is just a number, whereas A x B produces a new vector. So to calculate A · B you just have one formula, namely (2). To calculate A x B you actually have three formulas, one for each component, which go like
(A x B)_x = A_y B_z - A_z B_y (3),
etc. If you then calculate the magnitude of the new vector A x B, you will get formula (1) back.

So the most important ingredient to clearing up your confusion, I think, is that you should see formula (3) as defining the cross product, and see formula (1) for the magnitude as a consequence of that.