- the magnitude of a × b is the area of the parallelogram determined by a and b, and
- the direction of a × b is given by the right-hand rule .
That's all well and good, although I was at one point asked what good the cross product was. This is a tricky question to answer, because it is not immediately obvious to students of calculus why one would want to find a vector that is orthogonal to two given vectors; furthermore, the definition of the cross product is a lot more complicated and mystifying than the definition of the dot product, and I suspect they want something more obviously useful in return for that.
Anyway, I couldn't think of a geometric interpretation of the dot product, in the sense that this student seemed to want. The Wikipedia article says the following: Since |a|cos(θ) is the scalar projection of a onto b, the dot product can be understood geometrically as the product of this projection with the length of b. That's true, and I did say that, but what good is it? The two vectors that we're saying a ⋅ b is the product of the lengths of are in the same direction, so they don't naturally define an area. Is there a geometric interpretation of the dot product that doesn't feel so contrived? (This is sort of a vague question, as some people might not find what I just said contrived.)
We then moved on to the Scalar triple product a ⋅ (b × c), which created other frustration; it's weird to write this thing on the board and not mention that it's symmetric under even permutations of the arguments, because it's really the determinant of the 3-by-3 matrix containing those entries. I mumbled some words about how that sort of determinant comes up in changes of variables in multiple integrals, though; I was thinking of the Jacobian.
Of course, there's the classical claim that "you can only define a cross product in one, three, or seven dimensions", which I didn't mention, because nobody asked "can you define a vector product in two dimensions?" -- I would have mentioned it if they've asked.
But the cross product has that nice geometric interpretation. How do you continue with that? My officemate reminded me that the determinant of the "matrix"
(where i, j, k, l are the unit vectors in four-space) is the 3-volume of the parallelepiped spanned by the three vectors (a1, a2, a3, a4), (b1, b2, b3, b4), (c1, c2, c3, c4). So it seems that we can define some sort of "cross product" in this sense in any number of dimensions; in Rn it'll take n-1 arguments. This is actually the wedge product in disguise.
But then what's the "cross product" (in this sense) of a single vector in 2-space? It's not the vector itself; if you write the "determinant" it's
and so applying this operation to the vector (a1, a2) returns (a2, -a1). The operation takes in a vector and spits out its rotation by 90 degrees. This makes sense in retrospect; in the n-dimensional it spits out a vector which is orthogonal to the n-1 input vectors.
It just seems strange to be calling rotation by 90 degrees a "product"...