Find $k$ such that the vector with $w_n=1/(1+a_n k)$ is orthogonal to a given vector

Question

I am a little rusted. I have a vector $v$ with components $v_n$ in $R^N$, and another vector $w$ with components $w_n=\dfrac{1}{1+a_n k}$ in $R^N$. I have to find the value $k$ for which $v\cdot w=0$. Wich is the analytical solution?

Answer 1

So $\vec{v} = (v_k)_{k=1}^N$ and $\vec{w} = \left( \frac{1}{1+a_n k} \right)_{n=1}^N$. You must find $$ 0 = \vec{v} \cdot \vec{w} = \sum_{i=1}^N v_i w_i = \sum_{i=1}^N \frac{v_i}{1+a_i k}. $$ Can you continue from here? (Using numerical techniques, like Bisection or Newton's Methods, looks much easier unless $N$ is very small...)