I am trying to find a specific vector space as below. $v_0$ and $v_k$ are $n$-dimensional vectors, and for a given $v_0$ and $\theta$, the vector space is defined as: $V(v_0) = \{ v_k \in \mathcal{R}^n \, | \, <v_0,v_k> = |v_0||v_k|\cos(\theta) = 1 \}$.
How do I find the vector space $V(v_0)$? Can I somehow parameterize it in terms of $v_0$? I am trying to create a finite number of $v_k \in V(v_0)$ in Matlab. I would appreciate any help.
Could this work?
Assuming $|v_0| \cdot \cos \theta > 0$, \begin{align} &|v_0| \cdot |v_k| \cos \theta = 1 \\ \iff |v_k| &= \frac{1}{|v_0| \cos \theta} \\ \iff \sqrt{v_{k1}^2 + v_{k2}^2 + ... + v_{kn}^2} &= \frac{1}{|v_0| \cos \theta} \\ \iff v_{kn}^2 &= \frac{1}{|v_0|^2 \cos^2 \theta} - (v_{k1}^2 + v_{k2}^2 + ... + v_{k_{n-1}}^2) \end{align} You can pick all of the $v_{k1},\ ...,\ v_{k_{n-1}}$ freely, and then the final coordinate is determined by the above formula. This would produce a vector that must be in the set $V(v_0)$ by definition.