Finding vector coordinates given cosines and magnitudes

Question

I have three 3-dimensional vectors, $\boldsymbol{c_{1}} = \begin{bmatrix} c_{11*} \\\\ 0 \\\\ 0 \end{bmatrix}$, $\boldsymbol{c_{2}} = \begin{bmatrix} c_{21} \\\\ c_{22} \\\\ c_{23} \end{bmatrix}$, and $\boldsymbol{c_{3}} = \begin{bmatrix} c_{31} \\\\ c_{32} \\\\ c_{33} \end{bmatrix}$. If possible, I am trying to find the coordinates $c_{21}$, $c_{22}$, $c_{23}$, $c_{31}$, $c_{32}$, $c_{33}$. I know $c_{11*}$, as well all vector magnitudes and the cosines of the angles between the vectors. That is,

$||\boldsymbol{c_{1}}|| = R_{1} = \sqrt{c_{11*}^2}$, $||\boldsymbol{c_{2}}|| = R_{2}$, $||\boldsymbol{c_{3}}|| = R_{3}$,

$cos(\theta_{12}) = \rho_{12}$, $cos(\theta_{13}) = \rho_{13}$, $cos(\theta_{23}) = \rho_{23}$,

are all known.

I am able to find $c_{21}$ and $c_{31}$ by applying the dot product definition of the cosine:

$\boldsymbol{c_{1}} \cdot \boldsymbol{c_{2}} = cos(\theta_{12})||\boldsymbol{c_{1}}||||\boldsymbol{c_{2}}|| \implies c_{21} = \frac{R_{1}R_{2}\rho_{12}}{c_{11*}}$, and

$\boldsymbol{c_{1}} \cdot \boldsymbol{c_{3}} = cos(\theta_{13})||\boldsymbol{c_{1}}||||\boldsymbol{c_{3}}|| \implies c_{31} = \frac{R_{1}R_{3}\rho_{13}}{c_{11*}}$,

provided that $c_{11*} \neq 0$. Similarly, I know that:

$\boldsymbol{c_{2}} \cdot \boldsymbol{c_{3}} = cos(\theta_{23})||\boldsymbol{c_{2}}||||\boldsymbol{c_{3}}|| \implies c_{22}c_{32} + c_{23}c_{33} = cos(\theta_{23}) R_{2}R_{3} - c_{21}c_{31}$.

In addition,

$R_{2}+R_{3} = \sqrt{c_{21}^2 + c_{22}^2 + c_{23}^2} + \sqrt{c_{31}^2 + c_{32}^2 + c_{33}^2}$.

After that I'm kind of stuck. It seems that I still have 4 unknowns but only 2 equations. I tried understanding the problem in polar/(hyper)spherical coordinates, but that didn't help me either - quite the contrary, even.

So my questions are:

1. What am I missing and/or doing wrong?
2. Is there even a unique solution to my problem?
3. If there is a solution, does it generalize to any N-dimensional case (so to a case with N N-dimensional vectors instead of three 3-dimensional vectors)?

Answer 1

You can rotate the entire set of vectors around the axis $\boldsymbol{c_{1}}$ by any rotation angle without changing the length of any vector or the angle between any two vectors, and the inage of $\boldsymbol{c_{1}}$ will still be $\begin{bmatrix} c_{11*} & 0 & 0 \end{bmatrix}^T.$

This gives you a new set of vectors that satisfies all the defining properties you have given for your vectors (magnitudes and cosines of angles). This means you do not have a unique solution.

You can also reflect $\boldsymbol{c_{3}}$ through the plane that passes through $\boldsymbol{c_{2}}$ and $\boldsymbol{c_{1}}$ without affecting any of the defining properties.

You can get a solution by assuming that $c_{23} = 0$ and $c_{33} \geq 0.$ Then, if you like, you can parameterize the solutions by an angle of rotation about the axis $\boldsymbol{c_{1}}$ and then by a parameter whose value is $+1$ or $-1$ to represent whether to reflect $\boldsymbol{c_{3}}$ across the plane of the other two vectors.

I do not believe this gets better in higher dimensions.