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 $u = u_x \hat{x} + u_y \hat{y}+ u_z \hat{z}$ and $v = v_x\hat{x} + v_y \hat{y} + v_z \hat{z}$. The coefficients $u_i, v_i$ have units of length. You know the dot product

$u \cdot v = u_xv_x + u_yv_y + u_x v_z$,

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

$u \times v = (u_yv_z - u_zv_y)\hat{x} + (u_zv_x - u_xv_z)\hat{y} + (u_xv_y - u_yv_x)\hat{z}$

which is a vector. Its length is the magnitude of the parallelogram defined by $u$ and $v$. 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?