Angular momentum is traditionally taught as a (pseudo)vector quantity, tied closely to the cross product. This approach is familiar to experts but challenging for students, and full of subtleties. Here, we present an alternative pedagogical approach: angular momentum is described using bivectors which can be visualized as “tiles” with area and orientation, and whose components form an antisymmetric matrix. Although bivectors have historically been studied in specialized contexts like spacetime classification or geometric algebra, they are no more complicated to understand than cross products. The bivector language provides a more fundamental definition for rotational physics and opens the door to understanding rotations in relativity and in extra dimensions.

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Additional Appendices B-D and a sample class handout can be found in online supplementary material.
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Students accustomed to working with bivector tiles also have an especially easy time understanding the relationship between Kepler's second law and angular momentum, since the triangle formed by r and v d t is essentially just half of the = r p parallelogram.
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For a bivector defined in terms of the wedge product introduced in this section, it is straightforward to illustrate that changing the parallelogram's shape without changing its area leaves the result unchanged. For example, because the wedge product is linear in each argument, the bivector quantity 6 ( x ̂ y ̂ ) can be written as ( 6 x ̂ ) y ̂ , ( 2 x ̂ ) ( 3 y ̂ ) , ( 3 x ̂ y ̂ ) ( 2 y ̂ ), or infinitely many other forms, each corresponding to a differently shaped parallelogram with the same area.
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Intuitively, the idea behind this projection–rotation process is that the bivector is an area spanned by one vector sweeping around to a second vector. By taking a dot product with an additional vector from the left, we replace the first vector with a scalar, rotating the result toward the second. This can be made formal by reshaping the tile as a wedge product of a unit vector along the additional vector's projection and a second vector perpendicular to it. (A dot product from the right would rotate the resulting vector 90° opposite the bivector's orientation, reversing the sign.).
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Related to this, a rotation matrix can be constructed as the matrix exponential R ( t ) = e ω t. Then for a point with initial position r 0, the matrix product r ( t ) = R ( t ) · r 0 gives the rotated position as a function of time.
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One way of understanding this is that velocities in relativity do not add in a linear way: the Einstein velocity transformation means that different atoms of a rotating object will have their velocities changed by different amounts when boosted to a different frame, so Eq. (10) will no longer be true (even relative to the center of mass).
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If we do not demand that the two planes be mutually orthogonal, there are many other ways to write this final angular velocity as a sum of two wedge products. For example, [ ( 2 x ̂ + z ̂ ) ( y ̂ + 2 z ̂ ) + ( 2 y ̂ z ̂ ) ( y ̂ w ̂ ) ] rad / s.
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Supplementary Material

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