Scalars and Vectors: The Difference

[magu1re]

Hey. :)

I have just come onto working with vectors in pure mathematics and have no problems calculating with them. However, I do not really understand the difference between a vector and a scalar.

A scalar has magnitude only.

A vector has magnitude and direction.

Since trying to satisfy myself of the difference between these two objects, I have learned that scalars can be real numbers and are related to the operation of scaling a vector.

-7 is a scalar quantity althought it appears, to me at least, to be directed. I have been told that the negative is an indication of sense rather than direcition but this is a source of confusion for me. The magnitude of a scalar makes sense when I think of it as a vector. It seems to me as though it is a one-dimensional vector.

Would someone please clarify the difference between a scalar and a vector or provide any comments that could clear up this confusion?

Thank-you.

 

[Landau]

Yes, every scalar can be seen as a 1-dimensional vector.

Formally, scalars are elements of a field, say the reals R. A vector space over R is a group on which R acts; its elements are by definition called 'vectors'.

But R itself is a 1-dimensional vector space over R. In this sense, elements of R are both scalars (elements of the field R) and vectors (elements of the 1-dimensional vector space R over R).

(7 and -7 have opposite orientation. If we want to say that -7 has "negative direction", we are actually presuming an orientation on R. But that's a somewhat more advanced concept. )