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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 wv. But this "vector" does not start at the orgin  so is it actually the same as wv?)
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. :)
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 wv. But this "vector" does not start at the orgin  so is it actually the same as wv?)
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. :)
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