# Unicorns of Geometry: Getting Cross About Products

**Or: Why you should forget about the vector cross product!**

So titled because while both spoken about in tales and legends, neither unicorns nor cross products actually exist. Forget what you think you know about vector geometry!

A question for the beginner physics students: consider two vectors and . The coefficients have units of length. You know the *dot* *product*

,

which is a number, has units of area, and is the square of the length of one vector projected onto the other. You probably also know the other interesting operation, the *cross* *product*

which is a vector. Its length is the magnitude of the parallelogram defined by and . But its coefficients have units of area. How can it still be a vector in the same space as the other two if its coefficients have different units?