I am facing a linear algebra problem:
I want to calculate a (unit) vector $\ \overrightarrow{ AX }\ $ from a known point $A$ towards an unknown point $X$ in 3-dimensional Cartesian space.
Given are the vectors $\ \overrightarrow{ AB }\ $ and $\ \overrightarrow{ AC }\ $ as well as the angles $\ \angle BAX\ $ and $\ \angle CAX\ $.
I am aware that there are two points $X$ and $X'$ fullfilling the above mentioned conditions, that are mirror images with respect to $ABC$. However, retrieving either of those points would already help me a lot.
Thank you for your help!
Set: $$ \vec b={\overrightarrow{AB}\over|\overrightarrow{AB}|},\quad \vec c={\overrightarrow{AC}\over|\overrightarrow{AC}|},\quad \vec n = \vec b\times\vec c,\quad \angle BAX=\beta,\quad \angle CAX=\gamma,\quad \angle BAC=\alpha, $$ and write $\overrightarrow{AX}$ as a linear combination of base unit vectors: $$ \overrightarrow{AX}=r\vec b+s\vec c+t\vec n. $$ Taking the dot product of $\overrightarrow{AX}$ with $\vec b$, $\vec c$ and with itself we find the equations: $$ \cos\beta=r+s\cos\alpha\\ \cos\gamma=r\cos\alpha+s\\ 1=r^2+s^2+2rs\cos\alpha+t^2 $$ You can then easily find $r$ and $s$ from the first two equations and plug the result into the third equation to get $t^2$.