What is the difference between these two approaches?

[Vitani11]

Please take a look at this picture and tell me why using each of these approaches will give me different angles. When measuring the length (this is the outline of a razor) and using the equation to find the angle from the dot product, I get 53.6 degrees or so. When I just simply take the arc cosine of Q over P, I get 65 degrees or so. Why are these angles different? I don't have a protractor on me so I can't verify which angle it is. I'm positive that it is actually 53.6 degrees, because when taking the arc cosine of the components of the vector P I get 53.6 degrees which is the same as the angle found using the definition of dot product between Q and P. I know the angle is different with these approaches because the x-component for P is different than Q... but what's correct? Maybe my mind is going in circles.. Idk. I'm trying to internalize dot products and so I've been finding angles for polygons and triangles but I want to make sure I'm not being stupid.

 

[Vitani11]

Picture:

 

[Vitani11]

** I'm not asking you to calculate the answer as I'm aware I didn't give you any information - I just want to know which approach to use.

 

[Svein]

When I just simply take the arc cosine of Q over P, I get 65 degrees or so. Why are these angles different?

If you are going to use basic trigonometry, you need a right-angle triangle - and in this case you do not have one.

 

[Vitani11]

That is actually good to hear. I'm going to assume the latter equation is correct then since that is the calculus- based. So it is not cosine of Q over P but instead just the angle between the two vectors defined by the dot product? This approach worked beautifully for an irregular triangle I cut out of a piece of paper earlier, lol.