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I always wonder how vectors are used in real life.Vectors and decomposition of vectors,dot and cross products are taught in the early stage in every undergraduate physics course and in every university.My question is how and where are vectors used? Do physicists really use vectors in every day life? If so where? I'am Looking for Motivation for learning Vectors. |
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I am definitely not a physicist, however I can think of 2 engineering problems off the top of my head. If I'm incorrect about any of these, please let me know. 1)Structural engineering. If forces acting on structure are stronger than structure will support, the structure collapses. 2)Any kind of oscillator/wave propagation, including
Joe |
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Vectors are probably the most important tool to learn in all of physics and engineering. Some random examples:
To put it really simply, vectors are basically all about directions and magnitudes. These are critical in basically all situations. Force, Momentum and Velocity are all vectors. |
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Nobody actually uses undergraduate style vectors in real life, they are an inane useless outdated formalism which should not be taught. What people do use is the mathematically less sophisticated, but practically more useful, decomposition of vectors into components. The undergraduate calculus of cross products and dot products is incomplete, because it excludes operations which produce symmetric tensors, which show up all the time, and it is unweildy, because the cross product identities are counterintuitive. The real life formalism everybody uses is tensor index notation, as used and developed by Einstein and others at the turn of the 20th century. This notation replaces vector notation, is universal for tensors, and is directly reducible to component computations. When learning elementary physics, it is best to translate everything to index notation as quickly as possible. The history of vectors is William Rowan Hamilton's introduction of quaternions. Quaternions had a dot-product/cross-product multiplication, but they had 4 components. Physicists liked quaternions because they were mathematically elegant, and many papers used quanternions to express physical quantities. But in the 20th century, it became increasingly clear that quaternions were a peculiar algebraic structure which were useful for Lie groups, but had nothing to do with our three-dimensional space or our four dimensional spacetime. So physicists extracted the dot and cross product from the quaternion formalism, which butchered the whole scheme. The quaternions are a division algebra. Vectors with cross products are a nothing algebra. The quaternions are associative. Vectors under cross products aren't. All the elegance of quaternions was gone, and the clunkiness of the ill-fitting notation remained. All the operations of physics are better done by writing a vector as an object with indices, and manipulating the indices with contractions. This allows you to use tensors, which must not be kept hidden until graduate school. For an example of an impossible operation, consider the Navier Stokes equation $$ {\partial \over \partial t}{v} + (v\cdot \nabla) v + \nabla P + \nu \nabla^2 v = 0$$ Now take it's divergence. Oh no! You can't. Not in vector notation. You get $$ {\partial \over \partial t} {\nabla\cdot v} + \nabla \cdot (v \cdot \nabla) v + \nabla^2 P + \nu \nabla^2 \nabla\cdot v $$ Everything looks like it works, except when you try to expand the nonlinear term. You can't do it, because the nabla index is secretly contracted with the second v, and try as you might, there just isn't a vector expression which corresponds to the gradient of a vector field. It's a tensor. The various forms of Green and Stokes theorem are a nightmare in vector language, but they are trivial in indices. The cross product identities become trivial epsilon tensor identities, and you only use the epsilon tensor when your theory breaks parity, not all the time, like undergraduates are forced to do. For a recent example where vector notation makes it impossible to do a trivial calculation, see here: How did L.H. Thomas derive his 1927 expressions for an electron with an axis? |
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