Is there a way to recover the sum of a vector coefficients?

Question

Assuming an inner product between two vectors $\mathbb{a}$ and $\mathbb{b}$, $\langle \mathbb{a}\cdot \mathbb{b}\rangle$=v. Is there a way by knowing v and $\sum{\mathbb{b}}_i$ to obtain $\sum{\mathbb{a}}_i$

Answer 1

No, and the geometric reason is that if we let $x = \langle x_1,x_2,\ldots ,x_n\rangle$ be an arbitrary vector, $b$ a fixed vector, then $\langle b\cdot x\rangle = v$ actually cuts out a whole hyperplane in $\mathbb{R}^n$.

For example, $3x_1+3x_2 = 5$ is a line in $\mathbb{R}^2$, where $b=\langle 3,2\rangle$.

Thus, knowing $v$ and all the components $b_i$ won't allow us to pin down the second vector $a$.

I assume that is what you meant when you wrote $\sum b_i$, because if we only knew this sum and not each $b_i$, we actually know a lot less...