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It is only defined (and useful) in three-space, which makes it somewhat unusual (the dot product exists and is useful in any dimensional space).
The cross product is linear in its components: that is,
This means that we can define it on the unit vectors in three space, and then deduce it using the component-wise definition of a three-vector.
Now: here are the important geometric (rather than simply algebraic) properties of this product:
Example:
Examples:
#36, p. 692
I.e., creates a zero volume parallelpiped, because it's living in the plane of and .
This gives us our first equation of a plane: if the coordinates of are given by and the coordinates of are given by (that is is a normal vector) then the equation becomes
This is one form of the equation of a plane (through the origin). We say that the cross product is normal to the vectors that live in the plane (or normal to the plane): that is, that it is perpendicular to the plane.
Notice that we can instantly see a normal vector to the vectors in the plane in this form: just pick off the coefficients of the variables (a, b, and c) and turn those into a normal vector.
Now (for the second equation!): notice that satisfies the equation of the plane , since and are both perpendicular to their cross product. Hence we have another (parametric) equation of a plane (through the origin):
which leads to
where
which leads to