I am intrested in relationship between scalar and vector product in $\mathbb{R}^3$; I am going to give definitions which I will use in my question.
Scalar product – function $\cdot:\mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}$ which satisfay following properties:
a) $(\alpha \vec{u}+\beta\vec{v})\cdot \vec{w} = \alpha(\vec{u}\cdot \vec{w})+\beta(\vec{v}\cdot \vec{w})$
b) $\vec{u}\cdot\vec{v}=\vec{v}\cdot\vec{u}$
c) $\vec{u}\cdot \vec{u} \geq 0$ and $\vec{u}\cdot\vec{u}=0 \iff \vec{u}=\vec{0}$
Vector product – function $\times:\mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}^3$ which satisfay following properties:
a) $(\alpha \vec{u}+\beta\vec{v})\times \vec{w} = \alpha(\vec{u}\times\vec{w})+\beta(\vec{v}\times \vec{w})$
b) $\vec{u} \times \vec{v} = -\vec{v} \times \vec{u}$
c) $ (\vec{u} \times \vec{v}) \times \vec{w} +(\vec{v} \times \vec{w}) \times \vec{u}+(\vec{w} \times \vec{u}) \times \vec{v}=\vec{0}$
Question: We have endowed $\mathbb{R}^3$ with these two structures (structure of scalar product and structure of vector product). I know that space endowed with dot product is called inner space, is there a name for a vector space endowed with vector product? Are vector product and scalar product (as I have defined them) connected structures? Can we somehow exspress vecotor prodcut via scalar product or slacar product via vector prodcut? Is there some formula which can reduce on product to another? What preciasly is connection between these two structures, and can that connection be generalized to $\mathbb{R}^n$ (I do not know whether this question makes sense)?
Edit: I am hoping that if there is some connection between these two strcutures that answers not just say it, but rather explain why does this connection hold.
Thank you for any help.
Best Answer
You can easily define a scalar product in $\mathbb{R}^n$, for each $n\in\mathbb N$, be the vector product is specific to $\mathbb{R}^3$. There is a generalization to $\mathbb{R}^n$, but that's a map from $(\mathbb{R}^n)^{n-1}$ into $\mathbb{R}^n$.
A connection betwen both operations (assuming that you are dealing with the usual dot product here) is given by the triple product: given $v,w,u\in\mathbb{R}^3$, $\bigl\lvert v.(w\times u)\bigr\rvert$ is the volume of the parallelepiped defined by them.