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

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?

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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.
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.