A basic question about vectors

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In summary, the definition of vectors includes both magnitude and direction. They also satisfy the vector laws of addition, which are commutative, associative, and distributive. While this definition makes sense for many vector spaces, there are exceptions such as function vector spaces where adding and scaling functions is more abstract. It is also worth noting that current is not considered either a scalar or a vector, but it can be described as flowing from one point to another, indicating a direction.
  • #1
mantrapad
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I've found this regular definition of vectors: A quantity which has both magnitude and direction.

On some sites, I've also found the addition: They satisfy the vector laws of addition (commutative, associative, distributive).

Is it really necessary that all vectors should satisfy all those laws? Are there any exceptions?
 
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  • #2
mantrapad said:
I've found this regular definition of vectors: A quantity which has both magnitude and direction.

On some sites, I've also found the addition: They satisfy the vector laws of addition (commutative, associative, distributive).

Is it really necessary that all vectors should satisfy all those laws? Are there any exceptions?

Yes and no respectively. Otherwise it wouldn't be a vector!
There are other things that are similar to vectors, but we give them different names (like scalars and stuff).
 
  • #3
mantrapad said:
I've found this regular definition of vectors: A quantity which has both magnitude and direction.

On some sites, I've also found the addition: They satisfy the vector laws of addition (commutative, associative, distributive).

Is it really necessary that all vectors should satisfy all those laws? Are there any exceptions?


The "magnitude and direction" definition is an intuitive one. They make a lot of sense for R^n, but for some vector spaces, it doesn't. There are function vector spaces, where we create rules for adding and scaling functions. For example, if f(x) = x^2 and g(x) = x + 1, then (f + g)(x) = x^2 + x + 1. But what "direction" does f, g, or f+g point? It's much more abstract!
 
  • #4
Thanks for your replies and sorry about my late reply.

I was also wondering - current is neither a scalar nor a vector, right? But we say "current flows from A to B", which specifies a direction right? So it should be a vector...

And also, if it is neither, then which category does it fall in?
 

FAQ: A basic question about vectors

What are vectors and how are they different from scalars?

Vectors are mathematical objects that have both magnitude and direction. They are represented by arrows in a coordinate system. On the other hand, scalars only have magnitude and are represented by numerical values. For example, velocity is a vector quantity as it has both magnitude (speed) and direction, while time is a scalar as it only has magnitude.

How are vectors used in physics and other sciences?

Vectors are used extensively in various fields of science, including physics, engineering, and mathematics. They are used to represent physical quantities such as force, displacement, and velocity, and are essential for understanding and solving complex problems in these fields.

Can vectors be added or subtracted?

Yes, vectors can be added or subtracted to create a resultant vector. This is done by adding or subtracting the corresponding components of the vectors. The magnitude and direction of the resultant vector can be calculated using trigonometric functions.

What is the difference between a position vector and a displacement vector?

A position vector represents the location of a point in space relative to an origin point. It has a fixed magnitude and direction. On the other hand, a displacement vector represents the change in position of an object from its initial position to its final position. It can have varying magnitude and direction depending on the path taken.

How are vectors represented mathematically?

Vectors can be represented mathematically using coordinate notation, where the components of the vector are written as an ordered pair or triplet depending on the dimensionality. They can also be represented using vector notation, where the vector is denoted by a single bold letter or an arrow above the letter.

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