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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:
The "absolute values" in this case represent determinants, which, for a 2x2 system with first row a, b and second row c, d is given by . So the computation above works out to
Example:
Examples: