How do I solve to get vector from a dot product?

Question

Sorry if this question is quite elementary, I'm not very good at linear algebra. If the dot product of two vectors $\mathbf{a}\cdot\mathbf{b}=c$ then how do I solve for $\mathbf{a}$? I thought I could work it out by turning it into regular algebra ($c = a_1b_1+a_2b_2\ldots$). But after this step I'm stuck already. Please explain the answer if possible.

Answer 1

It should be pretty clear that there are infinitely many solutions $\mathbf a$, which means that we can't "solve for $\mathbf a$".

Let's look at an example in the $2$-D case: take $\mathbf b = (2,3)$, and consider $$ \mathbf a \cdot \mathbf b = 2 \implies 2a_1 + 3a_2 = 2 $$ Notice that this equation is solvable for any value of $a_2$, since we have $$ 2a_1 = 2 - 3a_2 \implies a_1 = 1 - \frac 32 a_2 $$ Which is to say we can consider $a_2$ to be a free variable. So, for example, $\mathbf a = (1,0),(- \frac 12, 1),(-2,2),(-5,4)$ are all solutions to this equation.