Vector Geometry and Vector Spaces

[SudanBlack]

Vector Geometry and Vector Spaces...

Hi - I've just started my degree course at university, studying theoretical physics. However, I have opted to attend the same maths lectures that some of the mathematics students are taking. We have been learning about "geometry and vectors in the plane", currently in R^2 space. The way we have defined vectors has their "tail" always at the orgin. (ie - a vector is an arrow pointing out of the origin) We have hence derived from this all of the necessary properties. (eg - we deal with addition of vecotrs by talking about parallograms, we have derived the scalar product using polar coordinates, etc)

However, when I was at school and indeed in my physics lectures, vectors do not always start at the origin. (for example, if vectors v and w both started at the orgin, in physics vector subtraction you would go from the "head" of v to the orgin to the head of w, forming w-v. But this "vector" does not start at the orgin - so is it actually the same as w-v?)

Obviously both methods must work, but since I was wondering if yhou could please explain to me how these 2 approaches are related? How are the mathematical principles I have been taught in my lectures extended to vectors not starting at the origin?

Many thanks in advance. :-)

 

[radou]

There is a difference between a vector and a radius vector. Vectors with 'tails' at the origin are called radius vectors, i.e. there is a bijection between R^2 and V^2(O). You can not talk about radius vectors unless you have defined a coordinate system.

However, when I was at school and indeed in my physics lectures, vectors do not always start at the origin. (for example, if vectors v and w both started at the orgin, in physics vector subtraction you would go from the "head" of v to the orgin to the head of w, forming w-v. But this "vector" does not start at the orgin - so is it actually the same as w-v?)

Think of w - v as of w + (-v). Apply the parallelogram rule. So, where does w - v 'start'?