Vectors and "displacement"

In summary, the conversation discusses the definitions of vectors as something with size and direction, and as "displacements in space." The concept of displacement as a vector itself and the representation of positions, forces, and other quantities by vectors with components is also mentioned. The conversation concludes that vector components represent the same quantity as the original vector and serve as a convenient abstraction for calculations.
  • #1
HyperActive
15
1
I'm just starting to learn about vectors, and I was trying to figure out what the vector components mean physically. I've seen two definitions of vectors, and the first is a that a vector is something with a size and a direction. The second definition I saw defined vectors as "displacements in space" and that given an example vector, say (3,4,1) that would represent a displacement of 3 in the x direction, 4 in the y direction and 1 in the z direction.

This made a lot of sense to me intuitively - that looking at it as displacement would explain why parallel vectors are equal and why vectors are coordinate-independent. However, although I don't know much about it, I know displacement is defined to be a vector in itself, so I can't see how it could be part of the definition of a vector.

So my question is, do vector components represent displacement? And if the don't, what physical quantity/term do they represent?

Thanks :)
 
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  • #2
Vector components are just other vectors, albeit ones which are usually parallel with a coordinate system or other convenient reference. Vectors can be used to represent positions in space, forces, moments, or other quantities which are composed of a magnitude and a direction.
 
  • #3
HyperActive said:
So my question is, do vector components represent displacement? And if the don't, what physical quantity/term do they represent?

Thanks :)
The way I read this, it seems the wrong way round. Displacement is an example of a Vector. A Vector is not (necessarily) a Displacement.
The components represent the same quantity as the original vector. The 'deeper' significance of what they mean is that they can be just an abstraction. There doesn't have to be a Force, Displacement or E Field in that particular direction. It's just convenient for the purpose of doing the calculation.
 
  • #4
Thank you both :) I think I understand now.
 
  • #5


Hello,

Thank you for your question about vectors and displacement. I can provide some clarification on this topic.

Vectors are mathematical quantities that have both magnitude (size) and direction. They are used to represent various physical quantities such as velocity, force, and displacement. In the context of displacement, vectors represent the change in position or location of an object.

The vector components, as you mentioned, represent the displacement in each coordinate direction. In your example, the vector (3,4,1) would represent a displacement of 3 units in the x direction, 4 units in the y direction, and 1 unit in the z direction. This is why parallel vectors have equal components, as they represent the same displacement in different coordinate directions.

Now, you are correct in saying that displacement is also a vector in itself. This is because it has both magnitude (the distance between the initial and final positions) and direction (the direction of the displacement). So, in a way, displacement can be thought of as the "ultimate" vector, as it is the combination of all the individual vector components.

To answer your question, yes, vector components do represent displacement. They show the magnitude and direction of the displacement in each coordinate direction. However, it is important to note that vectors can represent other physical quantities as well, such as velocity and force. It all depends on the context in which they are used.

I hope this helps clarify the concept of vectors and displacement for you. Keep exploring and learning about these concepts, as they are fundamental in many areas of science and mathematics.

Best of luck in your studies!

Sincerely,

 

1. What is a vector?

A vector is a mathematical quantity that has both magnitude (size) and direction.

2. What is displacement?

Displacement is the distance and direction an object moves from its initial position to its final position.

3. How is displacement different from distance?

Distance is the total length traveled by an object, while displacement takes into account the starting and ending points of an object's movement.

4. How are vectors represented?

Vectors can be represented by arrows, where the length of the arrow represents the magnitude and the direction of the arrow represents the direction.

5. What is the difference between scalar and vector quantities?

Scalar quantities only have magnitude, while vector quantities have both magnitude and direction.

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