Griffiths points out that not all twocomponent quantitities

Griffiths points out that not all two-component quantitities (the example is a \"vector\" with components representing apples and oranges) are real vectors. Why not? Write up your understanding of the argument for what distinguishes a vector in a few sentences

Solution

A \"vector\" with components representing apples and oranges are not real vectors, even though they do add like vectors. The reason is that this quantity does not tranform like other vectors (such as displacement) when the coordinate system is changed. The number of apples or oranges remain the same irrespective of the coordinate system used to describe its position.

Hence the correct definition of vectors is, \"a vector is any set of three components that transforms in the same manner as a displacement when you change coordinates.\" (Taken from Griffiths).

While trying to figure out if a quantity is vector or not we should always compare it with the way displacement vector transforms on change of coordinate system. If it transform like the displacement vector, then it is a vector else not.

Another interesting example worth mentioning is the electric current. It has a magnitude and a direction (in the sense we always define the direction of flow of current from high potential to low potential). The current flowing in opposite directions cancel each other. But still electric current is a scalar quantity (note: the current density is a vector quantity, current is not). Electric current does not transform like displacement vector on change of coordinate system. Also, electric current does not obey the law of vector addtion. Recollect the Kirchoff\'s current law, The sum of currents entering a node is equal to the sum of currents leaving the node. The current can enter or leave the node at arbitary angles, the law remains the same. This proves that current is a scalar quantity.