Find the minimum distance from the point $a=(a_1,a_2,...a_n)$ to the set $$X=\{ x=(x_1,x_2,...x_n)\mid b_1x_1+b_2x_2+...+b_nx_n=c \}$$
where $ b_1^2 + b_2^2 +...+ b_n^2 \gt 0 $ and $b_1, b_2,...,b_n,c \in \Bbb R$.
I have found so far the definition of distance from a point to a set in different books where stands that if (X,d) is a metric space $E\subset X$, $E\neq \varnothing$ and $x\in X$ we can define the distance from the point x to the set E in the following way
$$d(x,E):=Inf\{d(x,y): y \in E\}$$
but I have not found any example that could help me solve this problem, and I have no Idea of how to approach it
The method of Lagrange Multipliers tells you how to find the minimum value of $\sum\limits_{k=1}^{n} (x_i-a_i)^{2}$ subject to the condition $\sum\limits_{k=1}^{n} b_ix_i=c$. Consider the function $\sum\limits_{k=1}^{n} (x_i-a_i)^{2} -\lambda (\sum\limits_{k=1}^{n} b_ix_i-c)$ where $\lambda$ is a new parameter. The point where the minimum occurs can be found by setting derivatives w.r.t $x_1,x_2,\cdots,x_n,\lambda$ equal to $0$. This gives $2(x_k-a_k)-\lambda b_k=0$ for eaxh $k \leq n$. Solve this for $x_k$ in terms of $\lambda$ and then use the given condition $\sum\limits_{k=1}^{n} b_ix_i=c$ to find the value of $\lambda$.