In the order to a vector quantity, we say the quantity must have a magnitude and a certain direction only. But I think one more criteria should have been added is that, the quantity must have to follow the vector sum rule (like Triangle rule/parallelogram rule), i.e. if a quantity have a magnitude, and we can clearly associate or define a direction to this quantity, but this quantity do not follow the vector sum law (Triangle law), then it will certainly not be a vector quantity. For example electricity, it has a magnitude and we can associate a direction with it (say the conventional direction). Why this important Criteria (to follow the parallelogram law/triangle law) isn't mentioned in the definition of vector?
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$\begingroup$ You can have vectors in nonEuclidean space $\endgroup$– Carl WitthoftCommented Oct 8, 2021 at 14:19
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4$\begingroup$ No, the proper definition of a vector is that a vector is an element of a vector space. $\endgroup$– Vincent ThackerCommented Oct 8, 2021 at 14:22
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$\begingroup$ Yeah but when we google about vector quantity it shows only the two condition.Thats why i asked it $\endgroup$– MD HossainCommented Oct 8, 2021 at 14:55
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$\begingroup$ google.com/… $\endgroup$– MD HossainCommented Oct 8, 2021 at 14:57
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1$\begingroup$ @MDHossain Magnitude and direction is a very elementary (and narrow) definition of a vector. In addition, vectors only have magnitude and direction when an inner product (which is the dot product in $\mathbb{R}^n$) is defined. For a brief discussion on the parallelogram rule, see my answer here. $\endgroup$– Vincent ThackerCommented Oct 8, 2021 at 15:23
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In a Euclidean space, an additional condition is not really required. Displacement vectors clearly obey the triangle rule. The properties of all other vectors follow from that. Velocities and accelerations are derivatives from displacements. Momentum is mass times velocity. Force is mass times acceleration. Fields are defined in terms of force, etc. An electric current density is proportional to a resultant field (but there can be only one at a given point).
