How do we know for sure that vector addition works while working in 2D?Is there a mathematical proof for this?
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asked Feb 13, 2016 at 11:53
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3$\begingroup$ Do you mean Physicists?Could you just send me a link or write an answer please? $\endgroup$– The Cryptic CatCommented Feb 13, 2016 at 12:01
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$\begingroup$ I don't know what you are asking for. What do you mean "how do we know for sure that vector addition works"? What does it mean for addition to be "working"? How could it possibly fail to be "working"? $\endgroup$– ACuriousMind ♦Commented Feb 13, 2016 at 15:05
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$\begingroup$ math.stackexchange.com/questions/1407042/… $\endgroup$– Anubhav GoelCommented Feb 13, 2016 at 15:07
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$\begingroup$ m.youtube.com/watch?v=w8x8nETmD4w $\endgroup$– Anubhav GoelCommented Feb 13, 2016 at 15:10
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2$\begingroup$ The very definition of what constitutes a vector ensures that vector addition works. Go find someone who knows linear algebra and ask them "What is a vector?" $\endgroup$– Bill NCommented Feb 13, 2016 at 23:13
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1 Answer
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The parallogram rule for the addition of forces was described by Isaac Newton. You can test it with wires and calibrated spring scales in any undergraduate physics lab. Or use pucks on an air table.
For mathematical proof, see the first chapter of any linear algebra text for a full treatment.
The key is to know when a physical quantity is a vector: forces, velocities, accelerations, etc. Then you can rely on the math.
answered Feb 13, 2016 at 12:43
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$\begingroup$ Also, the usefulness of vectors depends on the validity of linear superposition. That is: the physical quantities being added must add linearly. This has to be verifies experimentally. There is no mathematical proof. In fact, it's not strictly true in some cases, such as the addition of electric fields in a medium. It's an approximation that works very very well in very many cases. But the entire field of nonlinear optics depends on it not being true. $\endgroup$– garypCommented Feb 13, 2016 at 13:13
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$\begingroup$ I agree with thus comment ... Maxwell's equations are linear in the fields for linear materials, hence superposion and the parallogram rule hold. But as intensities increase all materials will show weak nonlinear effects, and some will show strong nonlinearities. Thus the green laser pointer upshifts 1064 nm infrared light to 532 nm green light inside a crystal. I'm downconverting 402 nm UV light to 804 nm infrared light with a BBO crystal. This was first done in 1961 by Peter Franken et al at the University of Michigan. $\endgroup$ Commented Feb 13, 2016 at 13:21