# Finding vector coordinates given cosines and magnitudes

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:

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

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

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)?
• Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer.
– Community Bot
Nov 23, 2022 at 13:38

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 $$\begin{bmatrix} c_{11*} & 0 & 0 \end{bmatrix}^T.$$
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.