Assuming the exterior algebra $$ \left(\bigwedge\mathbb{R}^{n},\text{ }+,\text{ }\cdot,\text{ }\wedge\right) $$ exists and is well-defined, define the following operations on $$ \bigwedge\mathbb{R}^{n} $$: The interior product, the regressive product, the hodge dual, the scalar product and the geometric product.

Explore the operations in exterior algebra: interior product, regressive product, Hodge dual, scalar product, and geometric product. Gain insights into these advanced mathematical concepts with detailed explanations and examples.

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