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I know how to visually interpret single parts of a multivector. But what do they look like as a whole? Making an analogy with complex numbers doesn't work.

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You know that blades are directed measures. $a\wedge b$ for instance is a directed area, $a\wedge b\wedge c$ is a directed volume, so I suppose what you're asking is: what is for instance $1 + a + a\wedge b + a\wedge b\wedge c$?

It's a difficult question and I'm not sure anyone has a definite answer, but I suspect it's the same kind of question people often ask about a paravector $1 + a$ the first time they see one : "how can you add a scalar to a vector? That doesn't make any sense!"

Every time I'm confronted to this question I remember what a math teacher of mine told our class once in order to explain why scalar multiplication is noted $\lambda v$ and not $v\lambda$ : "people say I have two carrots, not I have carrots two". Ever since I've heard this I think of vector spaces as grocery stores. A vector is a bag in which you put stuff that can be very different. Apples and oranges are totally different things so you keep separate numbers to count them, but you can still put them in the same bag.

I hope that helps.

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Multivectors are mathematical abstractions - much like vectors, quaternions, and complex numbers. Certain applications may lend themselves to a geometric or visual interpretation of various component blades, but these are often the exception rather than the rule.

The formulation of geometric algebra allows us to use our spatial intuition when considering low-dimensional spaces (e.g. $d\leq 3$). This allows us to quickly learn the underlying operations of geometric algebra and gain confidence in their usage; however once this confidence is attained, the application of the mathematics is by no means limited to problems that have intrinsic geometric interpretations.

A vector can be used to describe positions, displacements or velocities, but it can also be used to describe tangents, normals, or any number of abstract quantities that can be described by the linear combination of a set of basis entities. Likewise, multivectors can be used to represent a wide variety of quantities that can be described as linear combinations of topological subspaces. Sometimes there is a convenient visual interpretation that can be used to gain additional insight into the problem, but often one must simply follow the math until a valid solution is obtained. The interpretation of that solution will then depend on the specific application and how its subspaces were encoded into the Geometric Algebra.

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  • $\begingroup$ You wrote "...multivectors can be used to represent a wide variety of quantities that can be described as linear combinations of topological subspaces. Sometimes there is a convenient visual interpretation that can be used to gain additional insight into the problem,..." Can you suggest a physical instance with a convenient visual interpretation that goes beyond the complex numbers and quaternions - say something that includes a trivector in a linear combination with a bivector or vector, describing a real physical situation, please? $\endgroup$
    – iSeeker
    Aug 10, 2022 at 15:12

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