Why are products of vectors just the way they are, and not the other way around?

Question

The scalar product and vector product of two vectors $\vec{A}$ and $\vec{B}$ are $AB\cos\theta$ and $AB\sin\theta\space\hat{\eta}$, respectively. They could have been the other way around—that is to say, $AB\sin\theta$ and $AB\cos\theta\space\hat{\eta}$, respectively—but why aren't they? Is there any particular significance behind the fact?

Answer 1

It comes down to the definition of the angle between vectors, as a scalar between $0$ and $\pi$. The scalar product gives you the length of the projection of one vector onto the other: $$\textrm{proj}_\vec{A}\vec B=B\cos\theta=\frac 1AAB\cos\theta =\frac{\vec A\cdot\vec B}{|A|}$$ The scalar product is positive if $\vec B$ has a component parallel to $\vec A$, negative if it has an antiparallel component, and $0$ if the two are perpendicular.

The vector product is used to describe a direction that is perpendicular to the plane of the two vectors. Now obviously, if the vectors are parallel or antiparallel, then they form a line, so there is no meaning to $\vec A\times\vec B$. Also or if one is $0$ you cannot define uniquely the perpendicular direction. All these cases are described by the $AB\sin\theta=0$ expression. That's why it was chosen for the vector product. It also has the advantage that this expression is the area of the parallelogram formed by the two vectors.

Answer 2

If you swap vectors, $\theta$ becomes $2\pi-\theta$. This preserves $\cos\theta$ but changes the sign of $\sin\theta$. On the other hand, their definitions tell us the dot product won't change sign but the cross product will.