Let A and B be two vectors in 3 dimensions.
Here seems to be the current treatment of the matter in Calculus textbooks:
If A = <x1, y1, z1> and B = <x2, y2, z2>, then the dot product is defined to be x1x2 + y1y2 + z1z2. Then, the distributive property and a few other useful properties are proven from the definition of the dot product. Next, a triangle consisting of sides A, B, and A-B is constructed. The law of cosines is applied to the angle between A and B. This finally yields the formula for calculating the cosine of that angle.
What I'm asking is basically, is there a rigorous definition of angles? Because, as you can see, the angle between A and B in the above discussion is just sort of taken to be a "de facto" entity.
I've imagined that one can use the differential equation y" + y = 0 to define the sine and cosine functions, prove the Pythagorean identity, and finally show that sine and cosine parametrize the unit circle. Once the unit circle is parametrized, I figure one can define angles between lines based on where they intersect the unit circle. Is there a simpler way to define angles? Or is that why all these introductory textbooks don't provide definitions?