| I think you hit the nailed on why most the textbooks are lousy at best on providing the 'why', apparently they are focusing more on the mechanical aspects and repeating exercises for scoring the exams, as a filler to the 1000 pages at US$100 textbook. One of the best books on Electronics according to HN crowd is The Art of Electronics, and it is filled with pages over pages of how-to of designing circuits of more than 1000 pages. But if you want to know why a Colpitts oscillator is the best for your design, all the best for that. Even the textbooks produced by professors from the best engineering schools (e.g MIT, Stanford, etc) are not spared of this issue. One of my former lecturers (not MIT) for linear algebra and numerical analysis courses claimed that he worked and consulted for NASA, but how I wished that he had cover some of the motivations of doing a dot product. For the ChatGPT responses of the reasons of doing dot product for two vectors see ChatGPT 4 prompt below. I think once ChatGPT/Gemini/etc can operate properly on math equations at least as good they can for NLP, I think we are going to have a wonderful treat even for the understanding the higly complex concepts for examples thermodynamics, electromagnetics and general relativity. Another classic example is try to find inside your favorite DSP book on why analytic signal in complex number generated by Hilbert Transform is necessary for communication and there is a very high probability that that's not a single explanation for that but there are many equations and graphs related to analytic signal. If we understand why or the raison d'etre, perhaps we can further ask why stop at analytic signal in complex number form, can we use quaternion number and what is the benefit of having more elaborate quaternion number form rather than complex form. If you are trying to do that and check this fact with most of DSP textbooks on this particular important issue then you will probably end up dissapointed even if you have bookshelf that contains majority of DSP books that are published since the invention of FFT. I have also included the ChatGPT prompt for the dot product, and the responses are delightful to say the least. >User prompt: Why you want to perform dot product of two vectors >ChatGPT 4: The dot product of two vectors, also known as the scalar product, is a fundamental operation in mathematics and physics with various applications, including: Computing angles between vectors: The dot product can be used to find the cosine of the angle between two vectors. This is particularly useful in geometry and 3D modeling to determine orientations and angles. Projecting vectors: The dot product is used in the projection of one vector onto another. This is crucial in physics for resolving a vector into components parallel and perpendicular to another vector, and in computer graphics for rendering and shading calculations. Checking orthogonality: Two vectors are orthogonal (i.e., the angle between them is 90 degrees) if their dot product is zero. This property is vital in many areas including computer graphics, signal processing, and linear algebra. Work calculation in physics: In physics, the work done by a force (a vector) moving an object through a displacement (another vector) is calculated using the dot product. This is important for understanding how much energy is being used or transferred in a system. Machine learning and data analysis: The dot product is used in algorithms such as those for finding similarities between data points, calculating vector norms, or in the training processes like those of neural networks where weights and input signals are combined. Each of these applications demonstrates why the dot product is a key tool in both theoretical and applied disciplines, linking geometry, algebra, and practical computation. |