Formality of Units in Cross Product

I recently lost some marks for answering a question on an assignment about the cross product of two lengths. I listed my answer in units of cm, instead of in units of cm^2 and that got me docked. This made me think a little bit about the units of cross product. I understand that in physical applications (like torque), we multiply together the units of the vectors which we are crossing. Algebraically, when we do the cross product, we multiply together components (each with units) and so the final output of the cross product has the products of units.

That’s totally fine with me, but it does feel very strange to take two vectors, multiply them together to get a new vector, and have that new vector have different units as the two vectors from which it is made. I’m perfectly happy with the cross product being thought of as the area of the parallelogram spanned by the two vectors being crossed, but I had always discounted the units. Having a vector with units of area feels weird.

In pure math applications we don’t really think of vectors as having units, but really they have units of “u”. So that means if I cross the î vector in the x direction with units (u) with the ĵ vector in the y direction (units u), I will get a k̂ vector in the z direction with units (u^2). Effectively this would mean that the z axis had different units from the x and y axes. This feels really weird, and is somewhat problematic because it means that the three statements: (î x ĵ = k̂) and (ĵ x k̂ = î) and (k̂ x î = ĵ) cannot all be true at the same time because the units of i, j, and k don’t work out.

Obviously, (î x ĵ = k̂) and (ĵ x k̂ = î) and (k̂ x î = ĵ) are all true statements, but I don’t understand how the units of cross product can be formalized while preserving important things like this.

Is it possible to think of the unit vectors î,ĵ, and k̂ as being unit less and then when expressing a vector in the vector space as some linear combination of those basis vectors with the coefficients on each vector containing the units?

Solution 1:

Since $(\vec{a}\times\vec{b})_i=\sum_{jk}\epsilon_{ijk}a_jb_k$ (coefficients here), if all $a_j$ have dimension $A$ and all $b_k$ dimension $B$ then all $(\vec{a}\times\vec{b})_i$ have dimension $AB$. Similarly, the dot product $\vec{a}\cdot\vec{b}=\sum_ia_ib_i$ has dimension $AB$.

You shouldn't be surprised by any of this, but I think I know why you are:

Having a vector with units of area feels weird.

This is what comes of asking students to visualize vectors as paths in the physical space we walk around in, forcing them to be of dimension length (i.e. $\mathsf{L}=\mathsf{L}^1\ne\mathsf{L}^2$). But like scalars, vectors can have any dimension. That's because a vector $\vec{v}$ is just $\sum_iv_i\vec{e}_i$ for dimensionless basis elements $\vec{e}_i$, with the coefficients $v_i$ being able to have any dimension (as long as they're all the same dimension; let's not get into an attempt to make mixed-dimension vectors work). There's no problem adding $v_1\vec{e}_1$ to $v_2\vec{e}_2$ in that situation.

Of course, when we multiply vectors with $\times$ or $\cdot$, or more generally multiply the components of one vector with those of another for whatever vector- or scalar-forming purpose, they needn't match, because it just reduces to the multiplication of scalars, which can happen with any dimensions for them.