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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?

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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.)

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  • $\begingroup$ 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? $\endgroup$ – Michael Stachowsky May 29 '17 at 1:48
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    $\begingroup$ 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? $\endgroup$ – Hans Lundmark May 29 '17 at 6:11

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