Geometrically the trivector a ∧ b ∧ c corresponds into the parallelepiped spanned by a, b, and c, with bivectors a ∧ b, b ∧ c and a ∧ c matching the parallelogram faces of your parallelepiped.
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This also pertains to the handedness from the cross product; the cross product transforms being a pseudovector beneath parity transformations and so is correctly referred to as a pseudovector.
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