Finding the solution to an equation where the left hand side is summation of orthonormal vectors

Question

I have this equation (I don't know what it's called):

$(\mathbf{u}_1+\mathbf{u}_2+\mathbf{u}_3)\cdot \begin{bmatrix}0\\1\\0\end{bmatrix} = c$

Where $\mathbf{u}_1$, $\mathbf{u}_2$, $\mathbf{u}_3$ are orthonormal. That is:

$\lVert \mathbf{u}_i \rVert = 1$ , and

$\forall i\neq j, \mathbf{u}_i\cdot \mathbf{u}_j=0$

Given $c$, how do I find the values of $\mathbf{u}_1$, $\mathbf{u}_2$, $\mathbf{u}_3$ that satisfy the equation?

Or more generally:

$(\sum\mathbf{u}_i)\cdot \mathbf{b} = c$

Given $\mathbf{b}$ and $c$, how do I find the values of $\mathbf{u}_i$ that satisfies the equation? Is there a name for this kind of problem?

I would like to know both analytical and numerical solution to this problem

Answer 1

Say $u_1 = (a, b, c), u_2 = (d, e, f)$ and $u_3 = (g, h, i)$. Notice that since we are making a scalar product with $(0,1,0)$, only the second entries of the $u_i$ will survive. So we have $$ b + e + h = c. $$ The above equation must be satisfied - as for the other entries of the vectors $u_i$, they can be anything in the field you're working over.

In the more general case, if $b$ is not as simple in the first case, then it's simply a problem of writing out the scalar product on the left hand side, equating to $c$ and solving for the entries of the $u_i$, though in the most general case there will be infinitely many solutions for the entries.

Answer 2

$u_1=(u_{11},u_{12},u_{13})$, $u_2=(u_{21},u_{22},u_{23}) $, $u_3=(u_{31},u_{32},u_{33})$

Then the solution for this equation:

$(u1+u2+u3)\cdot \begin{bmatrix}0\\1\\0\end{bmatrix}=c$ with: $\lVert \mathbf{u}_i \rVert = 1$ , and $ \mathbf{u}_i\cdot \mathbf{u}_j=0$

all $u_1$ $u_2$ $u_3$ such that: $$ \left\{ \begin{array}{lll} u_{12} + u_{22} +u_{32} =c \\ \sqrt{({u_{11}}^2 +u_{12}^2 +u_{13}^2)}=\sqrt{({u_{11}}^2 +u_{12}^2 +u_{13}^2)}=\sqrt{({u_{11}}^2 +u_{12}^2 +u_{13}^2)}=1 \\ u_i\cdot u_j=0 \space \space \forall i != j \end{array} \right. $$ With orthogonolity you come up to a finite number of solutions.