# Find a vector that satisfies the following: i) it has a given correlation with a second vector and ii) it is orthogonal to a set of vectors

I would like to generate a vector $$\vec{u}$$ of dimension $$n$$, so that i) it has a given correlation $$r$$ with a second vector $$\vec{v}$$ and ii) it is orthogonal to a set of $$m$$ vectors $$A = \{\vec{w}_1, \vec{w}_2, \dots, \vec{w}_k\}$$. I would like to do it in R. Any idea?

thanks a lot!

PS: Solving $$A \times x = 0$$ would fix ii). Also, satisfiying i) is relatively straightforward. However I have not been able to find a way to to ensure both i) and ii).

• Do you have any reason to believe that such a vector $\vec{u}$ can be guaranteed to exist? For example, if the given $\vec{v}$ happens to be in the span of $A$, then its correlation (I assume that you mean inner product) with any $\vec{u}$ is necessarily $0$. It is also difficult to see why this should not be a question of mathematics and only tangentially on topic in stats.SE. Mar 25, 2019 at 14:51
• @Dilip Because regression ideas are helpful in thinking about and solving this question, I think it is on topic here. User242367: what is the relationship between "$m$" and "$k$" in your question?
– whuber
Mar 25, 2019 at 18:05