cross product (thing)

Yet another cross product writeup? Well, everybody has his or her own outlook on things, and in this case I felt my outlook was sufficiently different to warrant another writeup.

The cross product is the complement of the dot product. While the dot product yields a scalar and the cross product yields a vector, this difference is largely semantical. In fact, it is reasonable to define a "scalar cross product" as the magnitude of the vector cross product. I will first discuss the scalar cross product, and then show how to find the direction of the vector cross product a X b.

The dot product of a and b is the projection of a on b times the magnitude of b. The scalar cross product is simply the component of a that is not projected onto b times the magnitude of b. See the 2-d diagram below.

              _ 
              /|
             / |
         a  /  |  component of 
           /   |  a not
          /    |  projected onto
         /     |  b
        / θ    |
       o========-------------->
        projection      b
        of a 
        onto b
        

As you can see, the scalar cross product is the complement of the dot product, though the determinant formula for cross products (given above) makes it look much more complex. Notice that while the dot product of a and b is given by a^.b = abcosθ, the scalar vector product is given by absinθ.

We've defined this "scalar cross product." What's the "real" cross product? The magnitude of the real cross product is the scalar cross product, and the direction is chosen such that it is perpendicular to both a and b.

There's just one more thing. There are actually two directions that will give perpendicular cross products. In the diagram above, one direction would be out of your monitor, and the other would be into your monitor. Mathematicians arbitrarily defined the direction of a X b to be into your monitor, and the direction of b X a to be out of your monitor. This is the so-called right hand rule. To find the "correct" cross product direction, point your right hand fingers along the vector on the left side of the X, squeeze them toward the vector on the right side of the X, and note where your thumb points.

Simple geometry/trigonometry shows that this definition of the cross product is equivalent to the determinant formula, which is useful in practice but isn't all that intuitive (to me at least).