Multiplying two Scalar dot products together stackexchange community,
I'm just wondering what the rules are for multiplying dot products together, such as:
$$
(P_{3}\cdot{P_{4}})(P_{1}\cdot{P_{2}})
$$
How would this be expanded out to not contain any dot products?
 A: Hint:
the scalar product of two vectors is a scalar, i.e. an element of the field over which is defined the vector space, so the product of two  scalar products is simply the product of two numbers in such a field.

Using linearity of the scalar product you can write:
$$
(P_3 \cdot P_4)(P_1 \cdot P_2)=\left((P_3 \cdot P_4)P_1 \right)\cdot P_2= P_1 \cdot \left((P_3 \cdot P_4)P_2 \right)
$$
A: The expression $(P_{3}\cdot{P_{4}})(P_{1}\cdot{P_{2}})$ defines a multilinear form, which is separately linear over each one of the variables $P_1,P_2,P_3,$ and $P_4$. More concretely, it is a degree-$4$ polynomial in the coordinates of $P_1$ through $P_4$. This polynomial can't be expressed as a sum of lower-degree polynomials, like the degree-$1$ polynomial $P_1+P_2$ or the degree-$3$ polynomial $P_{1}(P_{3}\cdot {P_{4}})+P_{2}(P_{3}\cdot {P_{4}})$.
If you like, you could hide the dot products behind Einstein notation: $\delta_{ij}\delta_{k\ell}P_3^iP_4^jP_1^kP_2^\ell$. Or, if the vectors are $3$-dimensional, you could probably turn the dot products into an elaborate dance of cross products. But one way or another, you're going to need some kind of multiplication operation, and lots of it, to build up that degree-$4$ polynomial.
