Summation of product identities

[eddybob123]

Hi everybody, I am just trying to find a decent identity that relates the sum $$\sum_{k=0}^{n}a_kb_k$$ to another sum such that $a_k$ and $b_k$ aren't together in the same one. If you don't know what I mean, feel free to ask. If you have an answer, please post it. Thanks in advance!

 

[tiny-tim]

hi eddybob123!

that's just a·b, the inner product (dot product) of two (n+1)-dimensional vectors a and b

 

[eddybob123]

But the a's and the b's aren't vectors, so what is the value of the ''$\cos\theta$"?

 

[mfb]

You can always interpret a and b as vectors.

There is no general formula to express this as the combination of two things, where one thing just depends on all a and the other thing just depends on all b.

In formulas, there are no (general) functions f,g,h to do this:
F=f(a_{0,a1,...,an)
G=g(b0,b1,...,bn)
h(F,G)=your result}

 

[Mandelbroth]

But the a's and the b's aren't vectors, so what is the value of the ''$\cos\theta$"?

The individual a's and b's might not be vectors, but we can think of them as components.

$$\sum_{k=0}^{n}a_{k}b_{k} = \begin{bmatrix} a_0 & a_1 & a_2 & \cdots \end{bmatrix}\begin{bmatrix} b_0 \\ b_1 \\ b_2 \\ \vdots \end{bmatrix} = \langle \vec{a},\vec{b} \rangle $$

How do we find the cosine of theta, you ask? $$\cos\theta = \frac{\displaystyle \sum_{k=0}^{n}a_{k}b_{k}}{(\sqrt{a_0^2 + a_1^2 + ...})(\sqrt{b_0^2 + b_1^2 + b_2^2 + ...})}$$


To be completely serious, I am not aware of the kind of general formula you are looking for.

 

[eddybob123]

And the magnitudes of the two vectors will be $$\sqrt{a_0{}^2+a_1{}^2+...+a_n{}^2}$$ and $$\sqrt{b_0{}^2+b_1{}^2+...+b_n{}^2}$$Is this right?

 

[tiny-tim]

yes