# Cross Product

by quantumfoam
Tags: cross, product
 Math Emeritus Sci Advisor Thanks PF Gold P: 38,705 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.
 P: 117 Ohhh. That makes a lot of sense! Is there anyway I could determine what the constant vector is?
P: 700
 Quote by quantumfoam 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...
P: 1,578
 Quote by micromass 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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