Wedge product intuition

Is the following interpretation correct:

Given two vectors in $\mathbb{R}^n$, their wedge product describes an oriented area with magnitude equal to the area of the parallelogram formed by the two vectors and orientation perpendicular to both?

Almost correct, except for “orientation perpendicular to both”.

If $n>3$, there are infinitely many directions perpendicular to the two vectors, so you can't think of the orientation as a vector (like you do for the cross product in three dimensions).

Instead, you may think of the orientation as a circle in the plane of the two given vectors $\mathbf{u}$ and $\mathbf{v}$, with a direction attached to it in one of the two possible ways: $\circlearrowleft$ or $\circlearrowright$.

The orientation corresponding to $\mathbf{u} \wedge \mathbf{v}$ is the direction in which you should rotate the first factor $\mathbf{u}$ to align it with the second factor $\mathbf{v}$. (The shortest possible rotation, of course, not the detour the other way around the circle.)

To align $\mathbf{v}$ with $\mathbf{u}$ instead, you must rotate in the opposite direction, so you get the opposite orientation, which algebraically corresponds to a change of sign: $\mathbf{v} \wedge \mathbf{u} = -\mathbf{u} \wedge \mathbf{v}$.

(For wedge products with more than two factors, it's not quite as simple. For $\mathbf{u} \wedge \mathbf{v} \wedge \mathbf{w}$, the two possible orientations correspond to whether $(\mathbf{u},\mathbf{v},\mathbf{w})$ form a right-handed or a left-handed system in the three-dimensional subspace of $\mathbb{R}^n$ that they span.)

• Is it correct to say that the orientation can be represented by a vector, which would be the axis of rotation? Or should I move away from vector ideas entirely? – Michael Stachowsky May 29 '17 at 1:48
• Rotations in $\mathbb{R}^n$ for $n>3$ don't have an axis of rotation (in general). For example, if $n=4$, if you rotate in the $x_1x_2$ plane, is the $x_3$ axis or the $x_4$ axis the axis of rotation? Or what about if you combine a rotation in the $x_1x_2$ plane with a rotation in the $x_3x_4$ plane? – Hans Lundmark May 29 '17 at 6:11