find vector involved in dot product when product and one vector is known

Question

suppose we have $2$ vectors $a$ and $b$ ($a$ is known and $b$ is unknown) and we have the dot product of $a$ and $b$. Is there any method to find the vector $b$?

eg:

$$a=[3,1,2] \\ b=[i,j,k]\\ a\cdot b=5. $$

find $i$,$j$ and $k$.

Answer 1

\begin{equation} \pmb{a}^{\text{T}} \pmb{b} = 3i + j + 2k = 5 \end{equation} This is an equation of 3 unknowns, therefore infinite number of solutions span 2 dimensions given as $\Big\lbrace (i,j,k) = (x,y,\frac{5-3x - y}{2})$ for all $(x,y) \in \mathbb{R}^2\Big\rbrace$.

Answer 2

Solve the linear equation $3i+j+2k=5$. The set of solutions of a single linear equation in $\mathbf R^n$ is an affine space of codimension $1$.