Cross Product of Constant and Vector

In summary: I'm sorry, I don't understand what you are saying.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...I'm sorry, I don't understand what you are saying.
  • #1
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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  • #2
The cross product is only defined between vectors of [itex]\mathbb{R}^3[/itex]. 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"
 
  • #3
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(:
 
  • #4
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?
 
  • #5
Sure! An equation like F=π[hXh+cXh] where h is a vector and c is a constant.
 
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  • #6
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.
 
  • #7
F is a vector.
 
  • #8
F=π[hXh+cXh] Sorry about not adding the equality.
 
  • #9
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 [itex]\mathbb{R}^3[/itex]. 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: [itex]B_{xy}, B_{yz}, B_{zx}[/itex]. The cross product can be thought of as the exterior product, combined with the identification of [itex]B_{xy}[/itex] with the unit vector [itex]\hat{z}[/itex], [itex]B_{yz}[/itex] with the unit vector [itex]\hat{x}[/itex], and [itex]B_{zx}[/itex] with the unit vector [itex]\hat{y}[/itex].

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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FAQ: Cross Product of Constant and Vector

What is the definition of the cross product of a constant and a vector?

The cross product of a constant and a vector is a mathematical operation that results in a new vector that is perpendicular to both the original constant and vector. It is denoted by the symbol "x" and is also known as the vector product.

How is the cross product of a constant and a vector calculated?

The cross product is calculated by taking the magnitude of the constant and the magnitude of the vector, multiplying them together, and then multiplying by the sine of the angle between the two vectors. This can be represented by the formula: A x B = |A| * |B| * sin(theta).

What are the properties of the cross product of a constant and a vector?

The cross product has several important properties, including: it is only defined in three-dimensional space, it is not commutative (meaning A x B is not equal to B x A), and it follows the right-hand rule (the resulting vector points in the direction of the curled fingers of your right hand when your thumb points in the direction of A and your index finger points in the direction of B).

What is the physical significance of the cross product of a constant and a vector?

The cross product has several important applications in physics, including calculating torque, angular momentum, and magnetic fields. It also has geometric applications, such as finding the area of a parallelogram formed by two vectors.

How is the cross product of a constant and a vector related to the dot product?

The dot product and cross product are both operations that involve two vectors, but they result in different values. The dot product results in a scalar (a number), while the cross product results in a vector. They are related through the distributive property, but they have different geometric interpretations and applications.

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