electricity - Law of vector addition

In one (Why is heat a scalar quantity?) of the questions I answered, I came across this:-

https://physics.stackexchange.com/a/404287/181020

or

A vector quantity should obey the law of vector addition.

But I don't think it is true. Consider a y junction circuit containing 3 identical wires. If i current flows through each of the 2 arms, the current through the other wire is 2i. But the current density vector of each arm doesn't add vectorially to give the current density in the third.

I want to know whether vectors should obey the law of vector addition or not.

Thanks in advance.

Answer

Vectors do follow vector addition; your Y-junction is not a counterexample to this, because you're applying the concept of "addition" to vectors in different places, which is not what is meant by "vector addition".

Vector addition for current fluxes means this:

• Suppose you run a current i through a wire


• Keep that current going, and send in another current i through the same wire


• Now the current through the wire is 2i

What your example shows is that the current flux is not the same everywhere in the circuit, but this is hardly surprising. If current flows around a loop of wire, the current flux vector points all different directions at different points, so of course it is not the same from place to place.