The function measuring the length of a line...isn't a linear function?

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Just trying to sort out some foundational issues here, when I noticed something really bizarre.

Say A(v^w) is a function which gives the area of v^w. I’ve reconciled myself to the area being able to be a negative number…and indeed, its desirable because you can do things like A(-v^w0 = -A(v,w), i.e. A is a linear function. All well and good.

But length??

Let L(v) be the function which assigned a length to the vector v. Under the euclidean metric, L(-v) does not equal -L(v).

I can’t believe I’ve been studying geometry for 30 years and never noticed this before. The length of a line…isn’t a linear function? Say what?

However, it seems like a “directed length” should make sense. And for basis vectors, there is an obvious convention: c*e1 has a positive length if cv is a positive scalar.

But what about the vector ce1-ce2? What sign should its length be? I just can’t seem to formulate any function L() which would be linear and would still preserve the intuitions I have about what a length of a vector should be.

SpaceTime Algebra metrics are even more puzzling… The dot product of a vector with itself can be negative…so we have imaginary lengths? And I thought negative area was bizarre. Nevertheless, would it help to formulate a linear function L() for the length if we threw those in?

I can’t sere how, but surely somebody else has thought along these lines—enlightenment would be appreciated!

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Actually, area is bilinear and not just linear. A(v,w) is linear in both v and w, making it bilinear.

Length of a vector is determined by a quadratic form, so yea it’s not linear. The quadratic equation comes from a bilinear operator, but the way this operates on vectors results in a quadratic polynomial instead of being linear.

There is such a thing as directed length in non-Euclidean spaces, where the signature of the quadratic form allows for negative values; however, in Euclidean spaces the sign of the length of all vectors is equal.

Typically, in spacetime geometry, the fact that the sign of the length changes is fundamentally tied to the fact that vectors are not purely in space dimensions and are not purely in time dimensions. Combination of space and time dimensions leads to a new type of geometry which is different than Euclidean spaces.

As you already figured out on your own, length is calculated from a quadratic formula, not a linear one.

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If I understand you correctly, the sign of signed area, then, is not analogous to directed length…because there’s no such thing as directed length.

If that’s true, please, everybody who is writing GA tutorials, please don’t try to build intuition about why areas are signed by referring to signed length.

In fact, this excavates a huge divide between the concepts of length and the concepts of N-volumes… they are two entirely separate concepts, one cannot be understood in terms of the other.

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When we refer to vectors, bivectors, etc. as directed entities, we don’t just mean that it has an extent in a given direction. It also has an orientation (e.g. from tail to tip). It helps if you avoid associating the sign with the scalar magnitude. Rather consider the sign and the scalar separately. The scalar denotes the magnitude of the quantity, and the sign denotes its orientation.

If the sign is positive, then the entity is aligned with the local unit basis entity. If it is negative, then it is oriented in the opposite direction. A similar intuition can than be applied to entities in any number of dimensions.

As for entities that square to negative scalar values; Once you square an entity, the resulting value is no longer of the same type (e.g. meters and square meters are describing two very different quantities). The scalar is proportional to the square of the magnitude of the entity (as established by the axioms of GA), however the sign is now describing something else. In this case, it relates to the behavior of the metric space in which the entity exists (i.e. Euclidean, Hyperbolic, etc.).

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Thanks for the reply! I guess where I’m still puzzled is that–even on the view you outline–the sense in which a vector has an orientation seems to be completely disjoint from the way a hypervolume gets its orientation.

Rather consider the sign and the scalar separately.

Well, the way the aforementioned signs and scalars are calculated is so radically different for vectors than it is for hyper volumes, that it seems to me that they should even have different names.

To emphasize the point, for the vanilla algebras which a newb would be familiar with, the sign of the length of a vector can never be less than zero. So we say that all non-zero vectors always have the same orientation? In vanilla algebras, even multiplying a vector by -1 won’t change its length. Surely this is completely different for hyper volumes?

And, as pointed out above, it’s computed with a quadratic function for a vector, which is ipso factor non-linear.

cf. with the orientation of a bivector or higher volume:

  1. They don’t all share the same orientation, like vectors (apparently?) mostly do.

  2. Multiplying a hypervolume by -1 will flip the orientation.

  3. And for hypervolumes, it is a bi- or multi-linear function.

I guess the most succinct way of putting my point is that for vectors, the length is calculated from the dot-product part of the geometric product, while volume is calculated by the wedge-product part of the geometric product. Completely disjoint parts of the geometric product—why is the same concept of orientation given to both? Seems like equivocation.

Once you square an entity, the resulting value is no longer of the same type

Absolutely. There is a very fundamental and deep distinction between length and volume. That’s why bivectors can’t be vectors (despite what that Yankee Dr. Gibbs says).

But this doesn’t help me understand why we lump the orientation of vectors in
conceptually with the orientation of columns: in fact, it just underscores how weird it is to do that.

I mean, this might seem like a small or pedantic point, but virtually every GA tutorial tries to build intuition for why volumes should be oriented by comparing/analogising/equivocating the orientation of vectors with the orientation of blades. This really tripped me up.

Seems like either:

  1. A newb reading such a tutorial would just bleep over the intro to orientation, in which case the comparison is doing no good, or,

  2. A newb like me, reading a tutorial like that, would pause to ponder it over a bit–and the more they thought about it the more mysterious it would appear. Points of preoccupation would be, in rough chronological order:

  • Hey wait, the magnitude of a vector is always positive (newbs haven’t learned about various different metrics yet) so how does it make sense even to say a vector is oriented?

  • But this tutorial says vectors have orientation, so what is the orientation of a vector? It can’t be the (positive) length of the vector…but then what the heck is it?

  • I can see how a bivector gets its orientation-- a^b = -b^a, So it would seem natural that the anti-commutativity of the wedge product would have some geometric meaning, And its straightforward that there are two ways–clockwise and counterclockwise–to sweep one vector to another. But if we are trying to do “the same thing” for vectors, we have a problem: there’s just one vector, what are we sweeping?

  • Hey wait, the function to calculate the length of a vector is not even linear, whereas calculating oriented hyper volumes is a linear function. The distinction between linear and non-linear in functions is a pretty big distinction, so why isn’t that distinction preserved in the terminology, like saying vectors don’t have orientation?

  • and (thanks to my conversational partner!) The units don’t even make sense. The length of a line is in meters, and the area of a bivector is in m^2.

Instead of a nice, smooth on ramp, the road just kind of breaks up into little chunks of concrete, and the further I go down it, the less of the road is left-- leaving me in the weeds :slight_smile:

Some conceptual help here would be very appreciated!! What is the orientation of a vector, and how do we compute it? And why does it make sense to lump:

  1. The square root of a dot-product, and
  2. The wedge product
    into the same category? Why would you use the same word, “orientation”, for both?