Let $x$ and $y$ be two vectors and $A$ the angle between them. Then we have the scalar product $$x\cdot y = \|x\|\|y\| \cos A$$
Let $x$, $y$ and $z$ be three vectors; $A$ angle between $x$ and $y$; $B$ angle between $x$ and $z$; and $C$ angle between $y$ and $z$. What is the value of the scalar product for the three vectors? Generalization: What is the value of the scalar product for $N$ vectors in $n$-dimensional space?
In 2-dimensional space we define a symmetric bilinear form for scalar product. In n-dimensional space can we define a symmetric multilinear form for N vectors?