I didn't use the template, because I am not having difficulties with a problem.

I am just starting to study rotational motion and there it appears the cross-product. I don't like to memorize formulae that I don't understand it's meaning.

Why is [tex]\vec a \times \vec b}[/tex] defined mathematically the way it is. Is there some trick to memorize?

If it helps, cross products are almost just like ordinary multiplication: if we write the standard basis vectors as i, j, and k, then all you need to know to compute a cross product is that

[tex]\begin{equation*}\begin{split}
i \times i = j \times j = k \times k = 0 \\
i \times j = k \\
j \times k = i \\
k \times i = j \\
j \times i = -k \\
k \times j = -i \\
i \times k = -j
\end{split}\end{equation*}[/tex]

(which is pretty easy to memorize), and that you can apply the distributive rule. (but not the associative rule, or the commutative rule!!!)

One geometric meaning to a cross products relates to perpendicularity -- you can already see that in the above identities. Another geometric meaning to the cross product of v and w is the "area" of the parallelogram with sides v and w, represented as a vector perpendicular to both v and w.

Arg, I remember asking this question a while ago. It is NOT "defined that way because it's useful".....grrrrr. There is a GEOMETRICAL PROOF for why it works.

Since [itex]\theta[/itex] is the azimuthal angle, then [itex]\bold{\hat{\theta}}[/itex] is the unit vector in the azimuthal direction. You can think of it in the same way as, say, [itex]\bold{\hat{x}}[/itex] is the unit vector in the direction of the x axis, then [itex]\bold{\hat{\theta}}[/itex] is the unit vector in the direction of the azimuthal "axis." Since [itex]\theta[/itex] is the azimuthal angle, the unit vector is thus the tangent vector in the direction of the azimuthal angle.