Why does a dot product equation have infinite (or many?) solutions?

Question

I am reading some books on geometric algebra (not the topic for the question), and one of them says that, if we want to solve for $\vec{x}$ given $\vec{a}$ and $\alpha$: $$\vec{a}\cdot\vec{x}=\alpha$$ Let any $\vec{x}$ be a solution, then any vector of the form $(\vec{x}+\vec{b})$ , $\vec{x}\cdot\vec{b}=0$ is also a solution.

I have tried playing around with equations and vectors, but I don't seem to get why. I only come up with cases where it doesn't work. Or perhaps I do not understand the question at all.

Here's the full extract, as well as the same question but with the cross product:

Thanks for any insights you may have and share.

Answer 1

It is just a matter of application of the linearity property of the inner product. Precisely, one has \begin{align*} \langle a, x + b\rangle = \langle a,x\rangle + \langle a,b\rangle = \langle a,x\rangle = \alpha \end{align*}

Hopefully this helps!