Cross Product of Constant and Vector

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Discussion Overview

The discussion revolves around the concept of the cross product involving a constant and a vector, exploring its definition, implications, and related mathematical expressions. Participants examine the validity of such operations within the context of vector mathematics and physics, including applications in magnetic fields and forces.

Discussion Character

  • Debate/contested
  • Mathematical reasoning
  • Conceptual clarification

Main Points Raised

  • Some participants assert that the cross product is only defined between vectors in \mathbb{R}^3, implying that a constant cannot be crossed with a vector.
  • Others introduce humor to highlight the impossibility of crossing a scalar with a vector.
  • One participant questions whether terms involving the cross product of a constant and a vector can be disregarded in equations, suggesting that it might be acceptable in certain contexts.
  • Another participant provides an example equation involving a vector and a constant, but others challenge its validity, stating that it does not make sense to include a cross product of a scalar and a vector.
  • Some participants discuss the nature of constants, with one suggesting that a constant can be a vector, leading to further inquiries about how to identify such a constant vector.
  • A later reply introduces the concept of the exterior product as a generalization, differentiating it from the cross product and explaining its implications in higher dimensions.

Areas of Agreement / Disagreement

Participants generally disagree on the treatment of constants in relation to vectors, with some asserting that constants can be vectors while others maintain that the cross product with a scalar is undefined. The discussion remains unresolved regarding the implications of these differing views.

Contextual Notes

Participants express uncertainty about the definitions and applications of constants and vectors, particularly in the context of magnetic fields and forces. There are unresolved mathematical steps and assumptions regarding the nature of constants in equations.

quantumfoam
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What is the cross product of a constant and a vector? I know that the cross product between two vectors is the area of the parallelogram those two vectors form. My intuition tells me that since a constant is not a vector, it would only be multiplying with a vector when in a cross product with one. Since the vector will only grow larger in magnitude, there would be zero area in the paralleogram formed because there is no paralleogram.
 
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The cross product is only defined between vectors of \mathbb{R}^3. The cross of a constant and a vector is not defined.

Lame Joke said:
"What do you get when you cross a mountain-climber with a mosquito?"
"Nothing: you can't cross a scaler with a vector"
 
So if I had an equation that contains a term that has a cross product of a constant and a vector, do I just cross it out of the equation? ( it is in an adding term so crossing it out would be okay). That's an awesome joke(:
 
quantumfoam said:
So if I had an equation that contains a term that has a cross product of a constant and a vector, do I just cross it out of the equation? ( it is in an adding term so crossing it out would be okay). That's an awesome joke(:

Can you give a specific example?
 
Sure! An equation like F=π[hXh+cXh] where h is a vector and c is a constant.
 
Last edited:
quantumfoam said:
Sure! An equation like π[hXh+cXh] where h is a vector and c is a constant.

That doesn't really make any sense.
 
F is a vector.
 
F=π[hXh+cXh] Sorry about not adding the equality.
 
Would the term containing the cross product of the constant c and vector h in the above equation just be zero? Or am I able to take cross it out of the above equation?
 
  • #10
quantumfoam said:
Would the term containing the cross product of the constant c and vector h in the above equation just be zero? Or am I able to take cross it out of the above equation?

No. As it stands, your equation makes no sense. You can't take the cross product of a scalar and a vector.
 
  • #11
Damn that stinks. Even if the c was a constant?
 
  • #12
quantumfoam said:
Damn that stinks. Even if the c was a constant?

Does this equation appear in some book or anything? Can you provide some more context?
 
  • #13
Well I made it up haha. I am sorry. I'm new at this. Do you think you can make an equation that makes sense? Like the one I attempted but failed at.
 
  • #14
quantumfoam said:
Well I made it up haha. I am sorry. I'm new at this. Do you think you can make an equation that makes sense? Like the one I attempted but failed at.

It only makes sense if you take the cross of a vector and a vector.

What were you attempting to do?? What lead you to this particular equation?
 
  • #15
Well, the h is a vector that represents a magnetic field strength. In the definition of a current, I=dq/dt, multiplying both sides by a small length ds would give the magnetic field produced my a moving charge. (dq/dt)ds turns into dq(ds/dt) which turns into vdq where dq is a small piece of charge and v is the velocity of the total charge. Integrating both sides to I ds=vdq would give the total magnetic field. For a constant velocity, the right side of the above equation turns into vq+ c, where c is some constant. Now I get the equation h=vq+c. Solving for qv gives me h-c=qv. In the equation for magnetic force on a moving charge, F=qvxB. I substituted h-c for qv in the above force equation. B turns into uh where u is the permeability of free space. I substitute uh for B in the magnetic force equation and get F=u[hxh-cxh]. I want the cxh term to go away.
 
  • #16
Does that sort of help?
 
  • #17
I don't understand any of what you said, but my physics is very bad. I'll move this to the physics section for you.
 
  • #18
Thank you very much!(:
 
  • #19
Saying that c is a 'constant' doesn't mean it is not a vector. A "constant" is simply something that does not change as some variable, perhaps time or a space variable, changes. In your formua c is a constant vector.
 
  • #20
Ohhh. That makes a lot of sense! Is there anyway I could determine what the constant vector is?
 
  • #21
quantumfoam said:
Ohhh. That makes a lot of sense! Is there anyway I could determine what the constant vector is?

A constant vector does not have to be a scalar ! A constant vector has a constant magnitude and a constant direction...
 
  • #22
micromass said:
The cross product is only defined between vectors of \mathbb{R}^3. The cross of a constant and a vector is not defined.

On the other hand, there is a generalization, the exterior product. The exterior product of a scalar and a vector is a vector. The exterior product of two vectors is a bivector. The exterior product of a vector with a bivector is a trivector. Etc.

In 3D, there are three independent bivectors: B_{xy}, B_{yz}, B_{zx}. The cross product can be thought of as the exterior product, combined with the identification of B_{xy} with the unit vector \hat{z}, B_{yz} with the unit vector \hat{x}, and B_{zx} with the unit vector \hat{y}.

Considering the result of the exterior product of two vectors to be another vector only works in 3D. In 2D, the exterior product of two vectors is a pseudo-scalar.
 
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