Vector Math: Adding, Subtracting & Division

In summary, vectors can only be added, subtracted, or “multiplied” by dot or cross product. This means that division of a vector by another vector is not allowed. However, you can divide the magnitude of a vector by the magnitude of another vector to get a scalar quantity, such as mass. There is no way to solve for mass using the vector form of Newton's second law, as it would require three equations for one unknown. The same applies for the vector form of the equation for torque. While complex numbers and quaternions allow for division, there is no practical use for vector division in these cases. Solving for a current given the force, field, and angle of the wire involves some sort of division,
  • #1
FS98
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As far as I know, vectors can only be added, subtracted, or “multiplied” by dot or cross product.

Does this mean that you couldn’t divide f-> by a-> to get m using the vector form of Newton’s second law? This would require dividing a vector by a vector, which seems to not be allowed.
 
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  • #2
Yes, you cannot divide a vector by another vector. Such division is meaningless. You can divide the magnitude of ##\vec F## by the magnitude of ##\vec a## to get a scalar quantity, namely the mass.
 
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  • #3
kuruman said:
Yes, you cannot divide a vector by another vector. Such division is meaningless. You can divide the magnitude of ##\vec F## by the magnitude of ##\vec a## to get a scalar quantity, namely the mass.
So can you not solve for mass with the vector form of Newton’s second law? Only the scalar form?

Or is there some way to get the answer by manipulating the vector form of the equation?
 
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  • #4
FS98 said:
Or is there some way to get the answer by manipulated the vector form of the equation?
Not really. A vector equation is three equations (one for each component). So solving it that way would be three equations for one unknown.

Basically, this type of operation won’t work on two arbitrary vectors, only on colinear vectors.
 
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  • #5
The great advantage that the Complex numbers have over vectors in R2 is that division by a non-zero element is always possible. The same can be said for the advantage of quaternions over vectors in R3.

Because division of complex numbers is defined, the derivative of a complex-valued function can be defined mimicking the definition of real-valued functions (limz->z0 ( f(z)-f(z0) )/(z-z0) ). The consequences are profound and beautiful.
 
  • #6
Dale said:
So solving it that way would be three equations for one unknown.
In the case of F = ma each of the three equations yields the same m. In the case of M=r x F you have no unique solution for r or F. So in either case there is no practical use for a vector division operation.
 
  • #7
A.T. said:
In the case of F = ma each of the three equations yields the same m. In the case of M=r x F you have no unique solution for r or F. So in either case there is no practical use for a vector division operation.
Unless you choose the direction of the unknown vector. So you can predict the necessary Current in a given Wire to produce a given Force in a Given Field. But that process has reduced things to a scalar operation I guess.
 
  • #8
sophiecentaur said:
...given Wire to produce a given Force in a Given Field.
That's 3 inputs, not a division operation on 2 vectors.
 
  • #9
But is shows the amplitude of one vector, given the other two and the direction you want. That involves division - it allows you to solve and equation which the bare fact that "you can't divide vectors" would seem to forbid.
 
  • #10
sophiecentaur said:
That involves division
Not of two vectors.

sophiecentaur said:
the bare fact that "you can't divide vectors"
You can define a "vector division" to be whatever you want it to be. Whether its useful or sensible to call it that, is another question.
 
  • #11
A.T. said:
You can define a "vector division" to be whatever you want it to be. Whether its useful or sensible to call it that, is another question.
To make a claim to the term "division", it should be the multiplicative inverse of some type of vector multiplication with a multiplicative identity. The dot and cross products do not have a multiplicative identity, so you would need to start with some new definition of vector multiplication.
 
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  • #12
A.T. said:
You can define a "vector division" to be whatever you want it to be. Whether its useful or sensible to call it that, is another question.
What would one call the process of finding the current if you knew the force and the field (plus the angle of the wire)? If it's not called division then what do you call it?
I understand that there is not an inverse operation for vector of scalar multiplication but the information can be found.
 
  • #13
sophiecentaur said:
What would one call the process of finding the current if you knew the force and the field (plus the angle of the wire)? If it's not called division then what do you call it?
As you stated earlier, you can just call it "solving for the current". The solution may involve some sort of division by non-vectors, but I would object to calling it vector division. The term "vector division" is just too strong to apply here.
I understand that there is not an inverse operation for vector of scalar multiplication but the information can be found.
The solution to a particular problem can be found, but that is not the same as having a guaranteed ability to divide by a non-zero vector in all situations.
 
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  • #14
@FactChecker Right. That sums it up nicely. Cheers. (A bit of personal interpretation is always better than a simple assertion of a fact.)
 
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Related to Vector Math: Adding, Subtracting & Division

1. What is vector math?

Vector math is a branch of mathematics that deals with the addition, subtraction, and division of vectors. Vectors are mathematical objects that have both magnitude and direction, and are often used to represent physical quantities such as force, velocity, and acceleration.

2. How do you add two vectors?

To add two vectors, you must first make sure that they have the same number of dimensions. Then, you add the corresponding components of each vector together to find the resultant vector. For example, if vector A has components (3, 5) and vector B has components (2, 7), their sum would be vector C with components (5, 12).

3. Can you subtract two vectors?

Yes, you can subtract two vectors in the same way that you add them. Instead of adding the corresponding components, you would subtract them. For example, if vector A has components (3, 5) and vector B has components (2, 7), their difference would be vector C with components (1, -2).

4. What is vector division?

Vector division is a mathematical operation that involves dividing a vector by a scalar (a single number). This is done by dividing each component of the vector by the scalar value. For example, if vector A has components (6, 12) and we divide it by 2, the resulting vector would be (3, 6).

5. How is vector math used in real life?

Vector math has many real-life applications, such as in physics, engineering, and computer graphics. It is used to calculate forces, velocities, and accelerations in physical systems, and to create realistic 3D graphics in video games and movies. It is also used in navigation systems and GPS technology to determine the direction and distance between two points.

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