Vector Geometry and Vector Spaces

In summary: It starts at the 'head' of w, but you can also think of it as starting at the 'head' of v. This is because the vector v is now 'translated' to start at the 'head' of w. This is how the mathematical principles of vectors can be extended to vectors that do not start at the origin. In summary, the concepts of vector geometry and vector spaces involve defining vectors with 'tails' at the origin, known as radius vectors. However, in physics, vectors may not always start at the origin. This can be accounted for by using the parallelogram rule and 'translating' vectors to start at a different point. This allows for the extension of mathematical principles to vectors not starting at
  • #1
SudanBlack
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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 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. :-)
 

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  • #2
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.

SudanBlack said:
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'?
 
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  • #3


Hi there! It's great that you're taking the initiative to attend math lectures that are outside of your degree program. It shows that you are eager to learn and expand your knowledge. Vector geometry and vector spaces are fundamental concepts in mathematics and physics, and it's important to have a solid understanding of them.

To answer your question, let's first define what a vector space is. A vector space is a mathematical structure that consists of a set of objects called vectors, and two operations - addition and scalar multiplication - that satisfy certain properties. These properties include closure, associativity, commutativity, and distributivity. The vectors in a vector space can be thought of as arrows, as you have learned in your lectures, but they don't necessarily have to start at the origin.

In fact, in a more general vector space, the vectors can start at any point in space. This is because the concept of a vector is independent of its position in space. A vector is defined by its magnitude and direction, not by its starting point. So, in physics, when we subtract vectors, w-v is indeed the same as v-w, regardless of where the vectors start.

Now, let's talk about vector geometry. In vector geometry, we are concerned with the geometric properties of vectors, such as length, direction, and angle between vectors. The principles you have learned in your lectures, such as using parallelograms to add vectors and using polar coordinates to calculate the scalar product, still apply in vector geometry, even if the vectors do not start at the origin.

The main difference is that in vector geometry, we are not limited to working in a two-dimensional space like R^2. We can work in higher dimensions, such as R^3 or even R^n. In these higher dimensions, it becomes more important to understand the geometric properties of vectors, as visualizing them becomes more difficult.

In summary, both vector geometry and vector spaces are important concepts in mathematics and physics. They are related in that vector geometry deals with the geometric properties of vectors, while vector spaces provide a mathematical structure for working with vectors. The principles you have learned in your lectures still apply, even when the vectors do not start at the origin, as long as they satisfy the properties of a vector space. I hope this helps clarify the relationship between vector geometry and vector spaces. Keep up the good work in your studies!
 

1. What is a vector in geometry?

A vector in geometry is a directed line segment that represents a magnitude and direction in space. It is typically denoted by an arrow above the letter representing the vector, such as v.

2. What is the difference between a vector and a scalar?

A vector has both magnitude and direction, while a scalar only has magnitude. Vectors can be added, subtracted, and multiplied by a scalar, but scalars cannot be added or subtracted. For example, displacement and velocity are vectors, while distance and speed are scalars.

3. How do you find the magnitude of a vector?

The magnitude of a vector is the length of the vector, which can be found using the Pythagorean theorem. If the vector is represented as v = (a, b), then the magnitude is given by |v| = √(a² + b²).

4. What is a vector space?

A vector space is a set of vectors that can be added and multiplied by a scalar, satisfying specific properties. These properties include closure under vector addition and scalar multiplication, associative and commutative properties of addition, and distributive properties of scalar multiplication over vector addition.

5. How do you determine if a set of vectors is linearly independent?

A set of vectors is linearly independent if none of the vectors in the set can be written as a linear combination of the others. In other words, if c₁v₁ + c₂v₂ + ... + cₙvₙ = 0 only when all the constants c₁, c₂, ..., cₙ are equal to 0. This means that none of the vectors can be "redundant" in the set, as they all contribute unique information to the space.

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