I know the cross product between a vector $a$ and a vector $b$ is just a vector whose magnitude is the product of magnitude of $b$ times the magnitude of the perpendicular component of $a$ in relation to $b$.This arises in many applications like in calculating the torque and the magnetic force.
The magnitude of this vectors definies how perpendicular $a$ and $b$ are … The more perpendicular, the higher the magnitude.This fact implies that, in order for it to be a measure of perpendicularity, we need the magnitude of $a \times b$ to be $\lvert a \rvert \lvert b \rvert \sin(\theta)$ and that should be a no-brainer.
It turns out that measuring the "perpendicularity" is the same as measuring the area between the paralelogram formed by the two vectors, the more perpendicularity, higher the area that is formed.
Concluding, I fully understand the reasoning between why the magnitude of the cross product is the way it is. But i want to keep building my intuition on Cross Product and I'm kinda stuck with two problems:
1- What is the reason (other than the fact that the magnetic force is vector perpendicular to both $v$ and $b$, and that the normal vector to a plane is a vector perpendicular to two vectors in the plane, for example) that this measure of perpendicularity ($\lvert a \rvert \lvert b \rvert \sin(\theta)$) was attributed to the magnitude of a vector? Why wasn't the cross product defined as just this magnitude? Was the orthogonal vector just some convenient form of killing two birds with a stone (getting both the measure of perpendicularity and getting the normal vector to the plane spanned by $a$ and $b$)?
2 – Is there any intuition that the components of cross product $a \times b$ are:
$ \langle(a_y b_z – a_z b_y), (a_z b_x – a_x b_z), (a_x b_y – a_y b_x) \rangle$? I'm saying intuition because I know the proof that the only vector (well one of the two possibles) who holds magnitude $\lvert a \rvert \lvert b \rvert \sin(\theta)$ and is perpendicular to both $a$ and $b$ should have this components.
Best Answer
Just a remark concerning 'problem 2':
The coordinates of the cross product $\bf{a}\times\bf{b}$ are the determinants of the projections of $\bf{a}$ and $\bf{b}$ onto the coordinate planes. So the $x$-coordinate of $\bf{a}\times\bf{b}$ is the area of the parallelogram spanned by the projections of $\bf{a}$ and $\bf{b}$ onto the $yz$-plane. I hope this helps your intuition a bit.