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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    $\begingroup$ To be honest, I don't know a single branch of physics that doesn't use vectors. $\endgroup$ – Michael Sep 18 '11 at 17:35
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    $\begingroup$ Vectors and Vector Spaces are simple, beautiful and as you can see in the answers useful. The position and all related quantities (velocity, momentum, acceleration, force) are vectors. The state of a physical system (both QM and classical) is a vector in a vector space. The solution of a system of (differential) linear equations is a vector. If you study physics, you will soon realize that most everything is a vector. $\endgroup$ – Jan Sep 18 '11 at 19:23
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    $\begingroup$ Do not learn vectors. Learn tensor calculus. Schutz's book on General Relativity is good for this. The undergraduate vector stuff is a useless nightmare, you only need to learn it to learn how to translate it to index notation. Mathematicians' vector spaces, on the other hand, are important. Learn abstract vector spaces. But these have nothing to do with what physicists call "vector calculus". $\endgroup$ – Ron Maimon Sep 19 '11 at 5:48
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    $\begingroup$ @alok , the answer you have accepted is not correct. Vectors (and the conventional notation) is used by many physicist in many different fields. Here are two random recent papers arxiv.org/abs/1106.2175 (optics) and arxiv.org/abs/0902.3952v1 (hydrodynamics). Notice that in the second paper, the authors use both vector and index notation, because they are convenient for different things. $\endgroup$ – Heidar Sep 20 '11 at 12:47
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    $\begingroup$ You can also see that most people which has responded, disagree with Ron. And if you want to learn physics, vector notation is essential to know, since most books are written in this notation (for example electromagnetism). $\endgroup$ – Heidar Sep 20 '11 at 12:50

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

    • AC electrical phase alignment/cancellation
    • sound/vibration propagation
    • RF propagation

Vectors are probably the most important tool to learn in all of physics and engineering. Some random examples:

  1. Classical Mechanics: Block sliding down a ramp: You need to calculate the force of gravity (a vector down), the normal force (a vector perpendicular to the ramp), and a friction force (a vector opposite the direction of motion).

  2. E&M: Electric fields and magnetic fiels are vector fields, with there properties determined in terms of vector calculus (Maxwell's Equations).

  3. Quantum Mechanics: In quantum mechanics you deal with infinite dimensional vector spaces (Hilbert spaces) as particle positions are unit vectors (functional spaces).

  4. Fluid Mechanics: In fluid mechanics, velocity in a pipe can be viewed in terms of a vector field.

  5. General Relativity: General Relativity is based in tensors, which are essentially generalization of vectors.

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.

  • $\begingroup$ He's not talking about abstract vector spaces. He's talking about the undergraduate vectors with arrows on them, the ones with the clunky cross product pseudo algebra. These are stupid and useless. $\endgroup$ – Ron Maimon Sep 18 '11 at 19:57
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    $\begingroup$ Well you need to understand vectors in order to understand vector spaces, and vector spaces are crucial in physics. In any event only (3) did I discuss vector spaces. I have no idea what you are calling "stupid and useless". $\endgroup$ – Benjamin Horowitz Sep 19 '11 at 1:26
  • $\begingroup$ The undergraduate style vectors with little arrows on top, with their calculus of dot and cross products, is useless. It is restricted to 3d, it is no good for calculations. Try to reproduce Thomas's precession using undergraduate vector calculus: physics.stackexchange.com/questions/14751/… $\endgroup$ – Ron Maimon Sep 19 '11 at 4:53
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    $\begingroup$ For the vast majority of undergraduate physics, standard vectors/vector notation work extremely well in communicating concepts and preforming calculations. Even if you are breaking them into components, the idea of it being "vector-like" still remains. $\endgroup$ – Benjamin Horowitz Sep 19 '11 at 5:17
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    $\begingroup$ I believe tensors have a time and place, after students gain a solid physical intuition. Anyone who is having trouble understanding basic vectors will definitely be completely lost with tensors. Also for those interested in engineering, tensors are usually overkill for most things. $\endgroup$ – Benjamin Horowitz Sep 19 '11 at 5:48

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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    $\begingroup$ Wait a second, I regularly use "undergraduate style vectors". For example, there is an elegant way of describing the track of a charged particle passing a giving surface - you just need the position on the surface (a 2-vector) and the direction of the particle at that point (a 3-vector). While we sometimes use vectors in the simple sense of an n-tuple of related numbers, what's really useful is the "physical" definition: An n-tuple with specific transformation properties, e.g. proper vs. axial vectors. There are a lot of deep things that can be formulated with those "little arrows". $\endgroup$ – jdm Sep 19 '11 at 12:56
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    $\begingroup$ I think you're taking a very black and white view here - vectors are a nightmare to work with, tensors aren't. Isn't it just down to knowing when to use one, and not the other? I find vectors incredibly elegant, especially when there are useful identities availiable for 3-vectors, not to mention the elegance of 4-vectors in special relativity. $\endgroup$ – Larry Harson Sep 19 '11 at 15:30
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    $\begingroup$ @Ron Maimon: that's not entirely true. Armed with a small list of vector [field] identities (for example the front flap of Griffiths' EM), you can sail through any (non-relativistic) EM or hydrodynamics class. Most physicists will probably use index notation to prove those identities, but that's another story. $\endgroup$ – Gerben Sep 20 '11 at 7:50
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    $\begingroup$ @4tnemele and Ron: I indeed learned those contractions with multiple dots, both for EM and for hydrodynamics. I'll be the first to admit that I haven't used them since, though. You can even do index-free GR: I believe it's done in 'General Relativity for Mathematicians' [Springer]. Again, I really don't want to get involved in the debate, as I believe that both abstract and component computations have their uses. $\endgroup$ – Gerben Sep 21 '11 at 21:37
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    $\begingroup$ @Gerben: The sophisticated methods are failures, other than Penrose's graphical index notation, because any attempt at an index free tensor notation, unless it is exactly equivalent to index notation, runs into the problem that exponentally many index-free expressions correspond to a given tensor expression. The multiple-dot symbols might be equivalent to indices, but they are not taught in undergraduate education (or anywhere else), and it does not change the fact that the calculus that is taught, undergraduate vector calculus, is incapable of expressing $\partial_j V_i\partial_i V_j$. $\endgroup$ – Ron Maimon Sep 21 '11 at 23:02

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