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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.

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    Have you learned about the dot/scalar product for vectors? – 2017-02-07
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    yes,I know some dot product and cross product – 2017-02-07
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    Euclid's Elements [book 1 proposition 18](http://aleph0.clarku.edu/~djoyce/elements/bookI/propI18.html): "*In any triangle the angle opposite the greater side is greater*". – 2017-02-07
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    @dxiv Thank you very much for your help, I am checking in my triangle – 2017-02-07

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You can construct vectors that connect your points. The vector coordinate will be equivalent to the difference of the responding endpoints coordinates.

$$\overrightarrow{AB}=(0, 60, 4);\quad\overrightarrow{AC}=(0, 80, -3);\quad\overrightarrow{BC}=(0, 20, -7)$$

The length of the sides is the same as the intensity of the vectors, which is equal to the square root of the sum of the squares of the coordinates.

$$|\overrightarrow{AB}\,|=\sqrt{3616};\quad|\overrightarrow{AC}\,|=\sqrt{6409};\quad|\overrightarrow{BC}\,|=\sqrt{449}$$

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.

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    Excellent explanation, with this method it is easier to find the greatest angle, many thanks – 2017-02-08