Why is time scalar, not vector?

[lowerlowerhk]

Let's see if I think correctly first: I think a vector is a group of numbers independent of each other. What we say 3D vector means "it takes three numbers to specify a position and these numbers are not (explicitly) dependent on each other. The so called 'direction' of a vector is a visualisation that reflects this property."

If the above is correct, and since time is also independent of spatial coordinates, then why can't time be a vector?
eg: in the definition of velocity as dx/dt, x is a vector while time is a scalar. Why so?

 

[Stephen Tashi]

You can think of a real number as an element of a one dimensional vector space. To formally make all the definitions and distinctions needed to define the one dimensional vector space is regarded as unnecessary bother unless the one dimensional space forms a subspace of a higher dimensional space.

 

[Prapti Bala]

Vector and scalar are physical quantities. And Vector has Magnitude and direction, satisfying the law of vector of addition. And Time doesn't have direction, this its scalar quality. And when a vector is multiplied, divided... with scalar, the quantity obtained is vector. So, when X, displacement is vector, when differentiating with time we obtain velocity,v which is also a vector quantity. Hope it helps...

 

[lowerlowerhk]

I got an answer, not sure if it is the complete answer:

The reason to not define time as another vector is that, in classical mechanics, the value of time is independent of reference frame. In math terms, it means that the value of time does not change under a coordinate transform and thus the length of the resultant vector magnitude might change. This defeats the very purpose of creating the concept of vector - to get rid of coordinate dependency.

In special relativity, where time does change under coordinate transform, time could be formulated as a component of a 4D space-time vector. This vector's magnitude is defined to be conserved under coordinate transform.