Suppose that $$A=\begin{pmatrix}\alpha_1 \vec{c} & \alpha_2 \vec{c}& \alpha_3 \vec{c}\end{pmatrix}, ~~\|\vec{c}\|=1, \vec{c}\in \mathbb{R}^3$$ Is fixed. Let $$f:O(3,\mathbb{R})\to \mathbb{R}, ~B\mapsto \mathrm{trace}(A^tB).$$ I want to find out the set $$\mathcal{S}=\left\{ B\in O(3,\mathbb{R}):~ B \text{ is global maxima of } f \right\}.$$
Edit Let me clear the problem by an example: If we take $2\times 2$ matrix and rank $1$ of the same type, i.e., $A=\begin{pmatrix}\alpha_1\vec{c}& \alpha_2\vec{c}\end{pmatrix}$, with $\|\vec|c\|=1,$ then there are exactly two matrices whose distance from $A$ is same. So basically I want to find out all the orthogonal matrix whose distance from the rank one matrix (in the above form, i.e., all the columns are non zero) is the same.
I hope the problem is clear, otherwise please comment so that I can clarify it more.