After trying to find some practical idea of how to find the largest angle of a triangle in 3D, I find myself in the painful situation of having no solution. The problem is the following:
If I have a triangle with vertices $A (0,0,7),\, B (0,60,11),\, C (0,80,4)$, how can I find out which vertex has the largest angle? I have not been able to find any method of calculating said vertex. If someone could guide me to be able to apply it to 3d triangles in general, it would be very helpful.
You can construct vectors that connect your points. The vector coordinates will be equivalent to the difference of the responding endpoint coordinates.
$$\overrightarrow{AB}=(0, 60, 4);\quad\overrightarrow{AC}=(0, 80, -3);\quad\overrightarrow{BC}=(0, 20, -7)$$
The length of the sides is equivalent to the intensity of the vectors. That is the square root of the sum of the squares of the coordinates (according to Pythagoras).
$$|\overrightarrow{AB}\,|=\sqrt{3616};\quad|\overrightarrow{AC}\,|=\sqrt{6409};\quad|\overrightarrow{BC}\,|=\sqrt{449}$$
It is evident that the largest side is $\overline{AC}$. The greatest angle is opposite to the largest side, therefore vertex $B$ has the greatest angle.