# Cross product as result of projections

The cross product between two vectors in $\Bbb{R}^3$ (call them a and b) is denoted a $\times$ b and the result is a vector in $\Bbb{R}^3$ orthogonal to the first two. There are a variety of ways of computing this resultant vector. One way in particular is known from the symbolic determinant involving i j k and the entries of a and b.

I recall a professor awhile back trying to explain why a $\times$ b can be seen from this by appealing to the notion of finding the projection of the parallelogram with sides a and b onto the xy, xz and yz planes (imagine a and b floating in 3-space, so we're finding the projection of a and b onto these planes and finding the new area of the parallelogram after this projection). The goal was to show that the factor scaling the i component of a $\times$ b was the area of the parallelogram formed by a and b after being projected onto the yz plane, and similarly for the j component being scaled by the area of the projected parallelogram of a and b in the xz plane and the k component from the xy plane.

The explanation seemed straightforward at the time, but in trying to remember it I'm a little lost. Is anyone familiar with this notion of finding the areas under projection and these areas being the magnitudes of the i j and k components of a $\times$ b?

I hope that isn't too vague. I know that we had an explanation for the sign change of the j component as well being connected to these smashed projections but I don't recall how that worked either.

• The monolith of words was a little intimidating, so I went ahead and used things called linebreaks to make it more likely that readers would read it all. Whitespace really does miracles for making text more readable! Nov 19, 2012 at 0:29
The magnitude of the cross-product $\|\mathbf{a}\times \mathbf{b}\|$ gives the area of the parallelogram formed by vectors $\mathbf{a}$ and $\mathbf{b}$ as edges. I do not know exactly what your professor is talking about so I will take a guess here.
From the Cauchy-Binet Formula, the area of the parallelogram is related to the areas of the projections of the parallelogram onto the $xy,\ xz$ and $yz$ planes. This is a higher dimension analogue of the familiar Pythagorean theorem. If we denote $\|\mathbf{a}\times \mathbf{b}\|_{xy}$ to be the area of the projection onto the $xy$-plane, then the areas are related as $$\|\mathbf{a}\times \mathbf{b}\|^2 = \|\mathbf{a}\times \mathbf{b}\|_{xy}^2 + \|\mathbf{a}\times \mathbf{b}\|_{xz}^2 + \|\mathbf{a}\times \mathbf{b}\|_{yz}^2$$ Now each projection is given by a smaller determinant. The $xy$ projection for example is given by $$\|\mathbf{a}\times \mathbf{b}\|^2_{xy} = \begin{vmatrix}a_1^2 + a_2^2 & a_1b_1 + a_2b_2 \\ a_1b_1 + a_2b_2 & b_1^2 + b_2^2\end{vmatrix}$$ where $a_i$ and $b_i$ are the respective components of $\mathbf{a}$ and $\mathbf{b}$. This is evaluated as $$(a_1^2 + a_2^2)(b_1^2 + b_2^2) - (a_1b_1 + a_2b_2)^2 = a_1^2b_2^2 -2a_1b_1a_2b_2+ a_2^2b_1^2 = (a_1b_2 - a_2b_1)^2$$ This is precisely the (square of the) third component of the cross-product.