# Vector Geometry

• Oct 7th 2006, 07:09 AM
SudanBlack
Vector Geometry
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. :-)
• Oct 7th 2006, 10:03 AM
topsquark
Quote:

Originally Posted by SudanBlack
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. :-)

First of all I congratulate you on choosing to take the Math course...I have yet to meet the Physicist that can adequately teach Math (including myself! ;) )

Basically the answer to your question is that you are working in an Euclidean space. The vector mathematics used to work with vectors in Physics at this level tends to ignore the "point of attachment" of the vector in question. So we are free to move the vectors around. You tend to lose this ability when you start working with non-Euclidean Mathematics in, say, Relativity. Particularly in General Relativity.

Oh, another example, and one that's more on the level you are speaking of. It is convenient to remember what point the vectors are attached to when you are working in rotational dynamics. For Newton's Law problems we don't care where the forces are located, but when we start talking about torques we need to know what point the forces are acting at.

-Dan
• Oct 7th 2006, 07:02 PM
SudanBlack
Thanks for the reply. :-) Could you possibly expand upon it a little bit for me please? You say in physics we ignore the "point of attachment" - is there an extension of the mathematics I am learning at university which allows vectors to no longer have start at the origin? Are we moving the vector or the axis?

Thanks. :-)
• Oct 8th 2006, 05:44 AM
topsquark
Quote:

Originally Posted by SudanBlack
Thanks for the reply. :-) Could you possibly expand upon it a little bit for me please? You say in physics we ignore the "point of attachment" - is there an extension of the mathematics I am learning at university which allows vectors to no longer have start at the origin? Are we moving the vector or the axis?

Thanks. :-)

For most of the applications in Physics that I have seen all we care about is the magnitude and direction of the vector, so we don't have to worry about where the vector is and we can move them around freely. For this purpose I suppose you would say that we are moving the vector, not the coordinate system. (Be warned: when doing rotations we usually consider the problem as rotating the vector, though there are applications where we would rotate the axes. The first is called an "active" rotation, the second a "passive" rotation, if I've got the terms right.)

I'm not sure exactly which Math class you would need to consider to find an extension of all this. It would be some version of a "tensor geometry" class or some such. The topics would include vector spaces, affine spaces, tensors, and connections. It would very likely be a graduate (or at least upper undergraduate) class so unless you are in grad school I wouldn't worry about such things for a while.

-Dan
• Oct 8th 2006, 10:31 AM
JakeD
Quote:

Originally Posted by SudanBlack
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. :-)

Hello. The pure mathematics of adding or substracting vectors in R^2 is simple. Your questions involve how to illustrate vectors and operations on vectors. These illustrations are outside the pure mathematics and are only aids to your understanding.

A vector in R^2 is a pair of numbers (x,y). Say in your graph, the vectors w = (5,5), v = (6,1). Then adding the vectors yields w+v = (5+6,5+1) = (11,6). And subtracting them yields w - v = (5-6,5-1) = (-1,4). That is all there is to the pure mathematics.

Notice there is no "head" or "tail" or "start" to a vector as a pair of numbers. What you are talking about is how to graphically illustrate the operations of vector addition and subtraction. In your graph, there are two ways to illustrate w-v: either as the red line connecting (-1,4) with (0,0), or as the dotted line connecting (5,5) with (6,1).

Both illustrations are useful so you don't have to choose between them. The purpose of both is to help you visually represent the operation of subtraction. You can use either as you prefer. There is nothing in the pure mathematics that distinguishes between these two methods of illustration.