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quantumfoam

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- Thread starter quantumfoam
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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.

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quantumfoam

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Physics news on Phys.org

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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"

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quantumfoam

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quantumfoam said:

Can you give a specific example?

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quantumfoam

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

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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.

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quantumfoam

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F is a vector.

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quantumfoam

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

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quantumfoam

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quantumfoam said:

No. As it stands, your equation makes no sense. You can't take the cross product of a scalar and a vector.

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quantumfoam

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

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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?

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quantumfoam

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quantumfoam said:

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?

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quantumfoam

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quantumfoam

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Does that sort of help?

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quantumfoam

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Thank you very much!(:

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HallsofIvy

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quantumfoam

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

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sankalpmittal

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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...

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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.

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.

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).

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).

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.

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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