Cross Product of Constant and Vector

[quantumfoam]

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.

 

[micromass]

The cross product is only defined between vectors of $\mathbb{R}^3$. The cross of a constant and a vector is not defined.

"What do you get when you cross a mountain-climber with a mosquito?"
"Nothing: you can't cross a scaler with a vector"

 

[quantumfoam]

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(:

 

[micromass]

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?

 

[quantumfoam]

Sure! An equation like F=π[hXh+cXh] where h is a vector and c is a constant.

 

[micromass]

Sure! An equation like π[hXh+cXh] where h is a vector and c is a constant.

That doesn't really make any sense.

 

[quantumfoam]

F is a vector.

 

[quantumfoam]

F=π[hXh+cXh] Sorry about not adding the equality.

 

[quantumfoam]

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?

 

[micromass]

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.

 

[quantumfoam]

Damn that stinks. Even if the c was a constant?

 

[micromass]

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?

 

[quantumfoam]

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.

 

[micromass]

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?

 

[quantumfoam]

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.

 

[quantumfoam]

Does that sort of help?

 

[micromass]

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.

 

[quantumfoam]

Thank you very much!(:

 

[HallsofIvy]

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.

 

[quantumfoam]

Ohhh. That makes a lot of sense! Is there anyway I could determine what the constant vector is?

 

[sankalpmittal]

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...

 

[stevendaryl]

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.