Why is vector addition defined the way it is?

In summary, I think that the definition of vector addition being based on the most natural way to combine directed line segments is due to the fact that vectors are a generalization of ordinary arithmetic and that they behave linearly in a lot of cases.
  • #1
Mr Davis 97
1,462
44
We all know that arithmetic addition arose not out of some axiomatic system, but out of the natural tendency to combine similar objects. I am satisfied with typical addition being loosely defined in this way. But when it comes to addition with other objects, like vectors, I am little bit confused. Why is that we define vector addition the way we do (i.e. a resultant vector is one that starts at the base of the first to the tip of the second)? Is this the definition of vector addition because it is the most "natural" way to "add" directed line-segments? Why does this definition happen to be so useful in physics?
 
Physics news on Phys.org
  • #2
Mr Davis 97 said:
Is this the definition of vector addition because it is the most "natural" way to "add" directed line-segments? Why does this definition happen to be so useful in physics?

The parallelogram addition of quantities like force have been known since antiquity (long before the concept of a vector was developed). Nature does not require that quantities add in this particular way, but it is really nice that they do! I would expect that the operation of vector addition is defined this way simply because the parallelogram addition of vectors has been experimentally verified.
 
  • #3
Vectors are generalizations of ordinary arithmetic. All finite dimensional vector spaces are isomorphic to either ## R^N ## or ## C^N ##, and to add vectors in either of these spaces, you add the components. This gives rise to the geometric construction for adding arrows, but the algebraic structure seems primary to me. Also, nature behaves linearly in so many instances, and when it doesn't, we find a way to linearize (usually). Scalar multiplication of vectors and vector addition are the essence of linearity. We break things up into pieces, operate on them with linear operators, and put the pieces back together again. Another addition rule, say a nonlinear one, would spoil our ability to do this.
 
  • #4
There are two different concepts here:
1) Mathematics. Given the definition of vector as element of a finite dimensional vectorial space, it is isomorphic to (e.g.) R^n as geofleur wrote, so that geometrical rule for summing vectors in space is easily proved summing the components of two vectors.
2) Physics. The fact some quantities as velocity, force, etc, are vectors it's an experimental fact and it doesn't follow mathematically from something else (as long as I know).

--
lightarrow
 
  • #5
Mr Davis 97 said:
not out of some axiomatic system, but out of the natural tendency to combine similar objects.
When you compare the very familiar arithmetical operations with the slightly less familiar vector operations we perform every day, you may be reading more into things than necessary. The formal processes with vectors may be a lot more complicated than simple arithmetical processes but we are constantly using vectors on an informal basis in our lives. The guy with a boat who's dong a 'ferry glide' doesn't do the sums but he can bring his boat perfectly to a stop at the jetty on the river by applying them. The darts player is allowing for the drop due to g, in every throw.
The original rules for arithmetic were not based on axioms any more than the rules used for using vectors. Mathematicians got hold of those rules and, with a touch of Post Hoc reasoning, commandeered them and turned them into an art form. Still, they gave us back more than they took so we can hardly complain.
What exactly is Mathematics? How basic is it to the operation of the Universe?
 

1. Why is vector addition defined as the sum of individual components?

Vector addition is defined as the sum of individual components because it allows us to break down a more complex vector into smaller, simpler parts. This makes it easier to perform mathematical operations and visualize the resulting vector.

2. Why is the commutative property applicable to vector addition?

The commutative property states that the order of operands does not affect the result. In the case of vector addition, this means that the order in which we add the vectors does not change the resulting vector. This property is applicable because the magnitude and direction of a vector are independent of its position in space.

3. Why is the parallelogram law used to define vector addition?

The parallelogram law states that if two vectors are represented by the adjacent sides of a parallelogram, then the diagonal of the parallelogram represents the resultant vector. This law is used to define vector addition because it allows us to visualize the magnitude and direction of the resulting vector when adding two vectors together.

4. Why is vector addition defined as the head-to-tail method?

The head-to-tail method involves placing the tail of one vector at the head of another vector to determine the resulting vector. This method is used to define vector addition because it allows us to easily visualize the resulting vector and determine its magnitude and direction.

5. Why is the magnitude of a vector determined using the Pythagorean theorem?

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In the case of a vector, the magnitude is the hypotenuse and the components are the other two sides. Therefore, the Pythagorean theorem is used to determine the magnitude of a vector because it accurately calculates the length of the resulting vector from its components.

Similar threads

Replies
4
Views
1K
  • Linear and Abstract Algebra
Replies
6
Views
710
  • Special and General Relativity
Replies
4
Views
1K
Replies
17
Views
344
  • Other Physics Topics
Replies
9
Views
16K
  • Special and General Relativity
2
Replies
38
Views
4K
Replies
23
Views
7K
  • Other Physics Topics
Replies
1
Views
8K
Replies
10
Views
608
  • Special and General Relativity
3
Replies
82
Views
5K
Back
Top