Given two sets of linearly independent vectors $\{v_1, v_2, v_3\}, \{w_1, w_2, w_3\}$ with each vector $ \in \mathbb{R}^{3}$, I am trying to find the linear combination with coefficients $\alpha_1, \alpha_2, \alpha_3 \in \mathbb{R}$ such that $$ \begin{align} r_1 &= \alpha_1 v_1 + \alpha_2 v_2 + \alpha_3 v_3,\\ r_2 &= \alpha_1 w_1 + \alpha_2 w_2 + \alpha_3 w_3,\\ \end{align} $$ yields orthonormal vectors $r_1, r_2 \in\mathbb{R}^{3}$, i.e. $$ \lVert r_1 \lVert^2 = r_1^\top r_1 = 1,\qquad \lVert r_2 \lVert^2 = r_2^\top r_2 = 1,\qquad r_1^\top r_2 = 0. $$
This problem can also be formulated as finding the vector $\alpha = (\alpha_1, \alpha_2, \alpha_3)^\top$ satisfying $$ \begin{align} \alpha^\top V^\top V\alpha = 1,\\ \alpha^\top W^\top W\alpha = 1,\\ \alpha^\top V^\top W\alpha = 0,\\ \end{align} $$ where $$ V = \begin{bmatrix} \vert & \vert & \vert\\ v_1 & v_2 & v_3\\ \vert & \vert & \vert\\ \end{bmatrix}, \qquad % W = \begin{bmatrix} \vert & \vert & \vert\\ w_1 & w_2 & w_3\\ \vert & \vert & \vert\\ \end{bmatrix}. $$
Does a solution to this problem exist? Any help is highly appreciated.