What is the difference between these two approaches?

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In summary, using the arc cosine of Q over P gives you a different angle than just taking the dot product between Q and P.
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Vitani11
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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.
 
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** 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.
 
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Vitani11 said:
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.
 
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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.
 

FAQ: What is the difference between these two approaches?

What is the difference between these two approaches?

The two approaches refer to different methods or strategies for achieving a certain goal or solving a problem.

What is the scientific approach?

The scientific approach is a systematic and logical process used by scientists to investigate and understand natural phenomena. It involves making observations, forming hypotheses, conducting experiments, and analyzing data to reach a conclusion.

What is the difference between the scientific approach and the empirical approach?

While the scientific approach follows a structured and controlled process, the empirical approach is based on direct observation and experience. The scientific approach relies on data and evidence to support or reject a hypothesis, while the empirical approach relies on personal observations and experiences.

Which approach is more reliable?

Both approaches have their strengths and weaknesses, and the reliability of each depends on the specific situation and research question. The scientific approach is generally considered to be more reliable because it is based on objective and systematic methods, while the empirical approach can be influenced by personal biases and subjective interpretations.

How do I know which approach to use?

The choice between the two approaches depends on the research question, available resources, and personal preferences. In some cases, a combination of both approaches may be used to achieve a more comprehensive understanding of a topic. It is important to carefully consider the strengths and limitations of each approach before deciding on the most suitable one.

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