I have a vector: $$\mathbf{v} = v_x \hat{x} + v_y \hat{y} + v_z \hat{z}$$
Which I know the components for ($ v_x , v_y, v_z$) for in the normal basis.
I also have the components of the vector in a new basis:
$$\mathbf{v} = v_1 \hat{e_1} + v_2 \hat{e_2} + v_3 \hat{e_3}$$
In this new basis, I know the components ($ v_1 , v_2, v_3$), but I do not know the basis vectors ($\hat{e_1},\hat{e_2},\hat{e_3}$).
I need to find the basis ($\hat{e_1},\hat{e_2},\hat{e_3}$) in terms of the normal basis - so this means finding the components:
\begin{bmatrix} \hat{e_1}\cdot \hat x & \hat{e_1}\cdot \hat y & \hat{e_1}\cdot \hat z \\ \hat{e_2}\cdot \hat x & \hat{e_2}\cdot \hat y & \hat{e_2}\cdot \hat z \\ \hat{e_3}\cdot \hat x & \hat{e_3}\cdot \hat y & \hat{e_3}\cdot \hat z \end{bmatrix}
However, I believe for the components I have specified there is many (perhaps an infinite number) of possible bases. So I have an additional constraint:
$$\hat{e_2}\cdot \mathbf{w} = 0 $$
Where $w$ is another vector which I know in the standard basis. i.e. I know the components ($w_x,w_y,w_z$).
I also have the constraint that the basis ($\hat{e_1},\hat{e_2},\hat{e_3}$) is orthogonal - i.e. $\hat{e_i}\cdot\hat{e_j} = \delta_{i,j}$.
So I need to know how to find the set of possible ($\hat{e_1},\hat{e_2},\hat{e_3}$) basis vectors, and from this set, how to apply these constraints to deduce the correct basis.
Please help!