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?
Asked
Active
Viewed 1.0k times
4
-
@GoodDeeds Could you show me for dot products? – Michael Roberts Feb 10 '16 at 17:43
-
Dot products have the distributive property. If you know the dot products separately, it's just like multiplying two scalars. – MathIsNice1729 Feb 10 '16 at 17:45
-
@SayantanSantra I don't know the two dot products separately, I was looking for an explicit answer... – Michael Roberts Feb 10 '16 at 17:51
-
1Are you asking how to write a dot product without writing a dot product? I'm confused... – bartgol Feb 10 '16 at 18:04
-
@bartgol Yes, I want to show something is the same as something else using the Mandelstam variables, I'm confused myself! – Michael Roberts Feb 10 '16 at 18:06
-
@MichaelRoberts It feels like you haven't asked your real question... – Chris Culter Feb 10 '16 at 18:20
-
@ChrisCulter Definitely not XYing the situation. – Michael Roberts Feb 10 '16 at 18:21
-
Please tell us what you know about the $P_i$'s. – MathIsNice1729 Feb 11 '16 at 03:11
2 Answers
4
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: $$ (P_3 \cdot P_4)(P_1 \cdot P_2)=\left((P_3 \cdot P_4)P_1 \right)\cdot P_2= P_1 \cdot \left((P_3 \cdot P_4)P_2 \right) $$
Emilio Novati
- 62,675
-
So would it be something such as $P_{1}\cdot{(P_3\cdot{P_{4}})}+P_{2}\cdot{(P_3\cdot{P_{4}})}$ – Michael Roberts Feb 10 '16 at 17:46
-
1No! $(P_3\cdot P_4)=p$ is a scalar ( a number, not a vector !) so $P_1 \cdot p$ is not defined ( the scalar product is defined only between two vectors). – Emilio Novati Feb 10 '16 at 17:53
-
-
-
Would there be a way to expand this out enitrely without the dot products? I'm really confused... – Michael Roberts Feb 10 '16 at 18:01
-
-
I'm not sure I follow. I just want the expression to be expanded out so there are no $\cdot$ terms, perhaps something like $P_{1}+P_{2}$, no dot products. – Michael Roberts Feb 10 '16 at 18:05
-
I think that you look for formulas similar to double cross product, but no such formulas exist for the double dot product. – Jean Marie Feb 10 '16 at 18:09
3
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.
Chris Culter
- 26,806