How do vectors relate to the area and length of a rectangle in vector calculus?

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Discussion Overview

The discussion revolves around the relationship between vectors and the area and length of a rectangle in the context of vector calculus. Participants explore the implications of the cross product of vectors, particularly how it relates to area, torque, and the dimensionality of vectors.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions how the equation C = A × B can equate a vector to an area, expressing confusion over the dimensionality of area versus length.
  • Another participant clarifies that vectors have magnitude and direction, and their units depend on the specific vector type, such as force or velocity.
  • Some participants discuss the utility of a vector representing area, suggesting it can be useful in defining concepts like the vector differential of surface area.
  • A participant introduces the idea that the cross product is technically a pseudovector, which raises questions about its properties compared to regular vectors.
  • There is a proposal of a new concept, the "ROCKY PRODUCT," which suggests an alternative way to define a parallelogram formed by two vectors, though it is noted that it would not satisfy existing vector calculus rules.
  • Participants discuss the significance of the direction of the torque vector, questioning whether torque can be applied at angles other than perpendicular to the plane formed by the vectors involved.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and agreement on the nature of vectors, the cross product, and the implications of these concepts in vector calculus. There is no consensus on the necessity or implications of the directionality of torque or the proposed "ROCKY PRODUCT."

Contextual Notes

Some participants express uncertainty regarding the dimensionality of vectors and the implications of the cross product, particularly in relation to torque and area. The discussion includes various interpretations of the significance of the pseudovector and its applications.

Who May Find This Useful

This discussion may be of interest to students and enthusiasts of vector calculus, physics, and mathematics, particularly those exploring the concepts of vectors, cross products, and their applications in different contexts.

rockyshephear
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I've been reading a bit on vector calculus trying to get a solid grasp on it. I am wondering how it can be stated that given C=AXB (all vectors), that C is equivalent to the area of the rectangle formed by A and B in the A,B plane. How can you equate length and area. One is in say...mm and the other would be in mm2. So are they saying that if A is 1 mm and B is 1 mm 90 degrees apart, the C is 1 mm3 perp to A and B? I thought vectors had only length and direction. Do you see my quandry here?
Thanks,
Rocky
 
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rockyshephear said:
I thought vectors had only length and direction.
They have magnitude and direction. They can be represented by "arrows" with length, but that length is merely proportional to the magnitude--the units of the magnitude depend on the particular vector being described. For example, force and velocity are both vectors, but neither is a length.
 
Thanks. So a cross product vector can represent velocity, acceleration, curl and volume or area among other things? I quess I don't know why I'd ever want a vector that is proportional to area in the first place.
Thx
 
Oh, I know what I was thinking. I was thinking that a vector can represent anyone dimenional quantity but it strains my thinking to see multi-dimensional quantities displayed by a vector.
 
rockyshephear said:
I quess I don't know why I'd ever want a vector that is proportional to area in the first place.
If the vectors A and B have units of length, then C = A X B will literally be a vector representing an area (units of length*length). But in general, the units of the cross product will be the product of the units of A and B. For example: Torque = r X F. r is a position vector with units of length and F is a force with units of force, thus torque has units of length*force. It's still useful to think of the cross-product as an "area" in a more general sense.
 
The "length" or "magnitude" of a vector is a number and numbers can represent many different things.

One reason we would want "a vector that is proportional to area" is to be able to define the "vector differential of surface area", d\vec{S}, normal to the surface and whose length is the differential of surface area.
 
Welcome to PF!

rockyshephear said:
… how it can be stated that given C=AXB (all vectors), that C is equivalent to the area of the rectangle formed by A and B in the A,B plane. How can you equate length and area.

Hi Rocky! Welcome to PF! :smile:

AxB is technically the dual of the 2-form (exterior product) A∧B …

we can also make 3-forms and so on (A∧B∧C∧…), and a k-form is k-dimensional (as one would expect!) …

the dual of a k-form is an (n-k)-form (where n is the dimension of the space)

and there's an isomorphism between all the k-forms and all their duals (the (n-k)-forms) …

in this case, n = 3, so the vector AxB is the dual of something that is two-dimensional. :smile:

(and just to make you even more confused … and I admit this has virtually nothing to do with your question :rolleyes: …

AxB technically is a pseudovector, not a vector (and (AxB).D is a pseudoscalar), since if you reflect in the origin, a vector comes out negative, but a pseudovector stays the same …

for that reason, we can never add vectors and pseudovectors. :wink:)
 
Thanks everyone. The torque example reallly cleared it up for me, I think. Here's my summary of what everyone said.

C=AXB (equation should be bold)
C is a pseudovector since you can't have a negative area
The units of C are the combination of the units of A and B as in dist and force in torque

So basically I am thinking that C is a kind of graph of what's going in the the plane that contains the two vectors A and B and creates the parallogram formed by them.

Just a crazy question: Since C is a sort of graph of the changes of the area of the parallelogram, is C a kind of integral of what is no longer an equation but a pair of vectors??
 
Why is the Cross Product needed in the first place? As a simplification, I would think that if one is presented with two vectors and asked to find the parallelogram formed, it's much simpler to just multiply the two magnitudes and have done with it. Just like in non-vector mathematics.
And why would the vector have to be normal to the plane that contains the two vectors? Is that arbitrary?
Can I now create a new vector calculus rule called the ROCKY PRODUCT that defines a paralellogram fromed by A and B in a plane, but the pseudovector is at 45 degrees to the plane holding the vectors. Or an infinite amount of new rules at differing angles?
Anyone see what I'm driving at?
Thanks
 
  • #10
the rocky product

Hi Rocky! :smile:

(are you the rocky product of your parents? :wink:)
rockyshephear said:
Can I now create a new vector calculus rule called the ROCKY PRODUCT that defines a paralellogram fromed by A and B in a plane, but the pseudovector is at 45 degrees to the plane holding the vectors. Or an infinite amount of new rules at differing angles?

You certainly can :smile: …

but it wouldn't satisfy certain very useful rules, such as:

(A+B)xC = AxC + BxC

and (AxB).C wouldn't be the volume of the paralellepiped (the wedgy thingy) :wink:
 
  • #11
You may be right but it may satisfy other yet unknown equations.
I think there is a world of as of yet untapped inbetween rules that will in the future all of a sudden become markedly useful. Call me the Nostrodamus of Vector Calculus but I believe that there is significance in currently insignificant correlations.

Here's a quick question. If AXB is the perp pseudovector that implies the area of the two vectors A and B in a certain plane for torque let's say. What is the significance of the normal vector C to torque? Cannot the torque magnitude plus the combined units of A and B, convey the same exact information about torque not matter what angle the pseudovector has in relation to the plane of A and B? for example. A=1 meter, B=1 Newton, so C=1 Newton/meter.
I would say that C's relative angle to A and B is insignificant.
Comments?
:)
Rocky
 
  • #12
rockyshephear said:
Call me the Nostrodamus of Vector Calculus …

Hi Nostrodamus of Vector Calculus! :biggrin:
If AXB is the perp pseudovector that implies the area of the two vectors A and B in a certain plane for torque let's say. What is the significance of the normal vector C to torque? Cannot the torque magnitude plus the combined units of A and B, convey the same exact information about torque not matter what angle the pseudovector has in relation to the plane of A and B? for example. A=1 meter, B=1 Newton, so C=1 Newton/meter.
I would say that C's relative angle to A and B is insignificant.

(btw, i wouldn't go around calling it a pseudovector, or people will think you're odd :redface:)

The significance is that torque is a vector (pseudovector) …

torque isn't just a number, you have to specify the direction (axis) of it …

the basic torque equation is torque equals rate of change of angular momentum, and both torque and angular momentum are vectors (pseudovectors). :smile:
 
  • #13
I will buy that but the cross product seems to imply that torque would neccessarily be perpendicular to the plane of A and B by definition. That's highly unlikely that all torque is applied in that direction so what about torque that is applied at an angle NOT perp to the plane of A and B?
The Karnac of Vector Calculus (downgrade lol)
 
  • #14
rockyshephear said:
The Karnac of Vector Calculus (downgrade lol)

Nostradamus would have seen that downgrade coming! :smile:
I will buy that but the cross product seems to imply that torque would neccessarily be perpendicular to the plane of A and B by definition. That's highly unlikely that all torque is applied in that direction so what about torque that is applied at an angle NOT perp to the plane of A and B?

Nope, you've lost me :redface: …

the A and B of torque are the force and the distance to the axis …

with that A and B, torque has to be perpendicular to them both.
 
  • #15
rockyshephear said:
Why is the Cross Product needed in the first place? As a simplification, I would think that if one is presented with two vectors and asked to find the parallelogram formed, it's much simpler to just multiply the two magnitudes and have done with it. Just like in non-vector mathematics.
"Have done with it" and just ignore the fact that it gives the wrong answer? The area of a parallelogram is the product of the two magnitudes only if the parallelogram happens to be a rectangle. The area of parallelogram is |u||v| sin(\theta) where \theta is the angle between the sides, exactly the formula for magnitude of the cross product of two vectors.

And why would the vector have to be normal to the plane that contains the two vectors? Is that arbitrary?
I answered that before. Distinguishing the normal to a surface is an important ability in applications and that is an important use of cross product. If a surface is given in terms of two parameters, u and v, with "position vector", \vec{r}(u,v), then the derivatives, \vec{r}_u and \vec{r}_v, are two vectors in the tangent plane to the surface. The "fundamental vector product", \vec{r}_u\times\vec{r}_v is a vector normal to the surface. It's magnitude, \left|\vec{r}_u\times\vec{r}_v\right| times "dudv", \left|\vec{r}_u\times\vec{r}_v\right|dudv is the "differential of surface area". To integrate a real-valued function over the surface, you write it as \int\int f(u,v)\left|\vec{r}_u\times\vec{r}_v\right| dudv. Even simpler is integrating an vector valued function over a surface (as for example, finding the flow through a surface or the flux of a magnetic field through the surface). That is just \int\int \vec{f}(u,v)\cdot\vec{r}_u\times\vec{r}_v dudv

Can I now create a new vector calculus rule called the ROCKY PRODUCT that defines a paralellogram fromed by A and B in a plane, but the pseudovector is at 45 degrees to the plane holding the vectors. Or an infinite amount of new rules at differing angles?
Anyone see what I'm driving at?
Thanks
Of course you can! Is it useful? I can think of many situations in which it is useful to separate a vector into components along and perpendicular to a surface: only the component of force perpendicular to a surface causes pressure on the surface, only the component of force along a surface will move an object over the surface. Can you give examples where a component at 45 degrees to a surface or at some other specific angle would be useful?

Apropos your post on "torque", you don't see to understand that torque is always measured relative to some axis. And is always perpendicular to that axis. Perhaps instead of railing against things you don't understand, you should learn them.
 
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  • #16
Railing against things in a gentile way is the best method for me to understand things I do not understand. It is rather like an exclamation point at the end of a sentence for executing emphasis. :)
As far as the torque analogy to the cross product, I'm still not grasping it properly. Could the entire thing be put into a non-mathematical statement for easier digestion? ie

The cross product vector (pseudovector), which is always normal to A and B, relates to the the axis of rotation around which the torque is applied and the magnitude of the cross product vector (pseudovector) is equal to the magnitude of the torque applied in whatever units the vector A and B are set to.

If that's pretty close to being accurate, I would still ask this really crazy question.

I would argue that the area of a parallogram is not A time B sin theta but simply A times B.

Here's my rational.

Vector A has a magnitude and B has a magnitude. Imagine a parallogram. Hold the bottom two points in place and rotate the top two points such that they line up with the bottom two points in the vertical axis. Now you have the same magnitudes just at different angles. And the area is not the actual magnitude's products. So you are saying that by shifting the top two points, you are actually changing the area inside the geometric object? It seems counter-intuitive. If you can explain how it does change the area, then I will believe it.

Thanks.
 
  • #17
Go the other way. Rotate the top two points so that they are in the same line as the bottom two points. Now the area is zero. I conclude that the area of any parallelogram is zero? Or I conclude that the area is not AB

Calculating the area of a parallelogram is fairly easy and straightforward; why don't you try to do it?
 
  • #18
rockyshephear said:
Railing against things in a gentile way
As opposed to a Jewish way?:-p

is the best method for me to understand things I do not understand. It is rather like an exclamation point at the end of a sentence for executing emphasis. :)
As far as the torque analogy to the cross product, I'm still not grasping it properly. Could the entire thing be put into a non-mathematical statement for easier digestion? ie

The cross product vector (pseudovector), which is always normal to A and B, relates to the the axis of rotation around which the torque is applied and the magnitude of the cross product vector (pseudovector) is equal to the magnitude of the torque applied in whatever units the vector A and B are set to.

If that's pretty close to being accurate, I would still ask this really crazy question.

I would argue that the area of a parallogram is not A time B sin theta but simply A times B.
You can "argue" that 2+ 2= 5 if you like.

Here's my rational.

Vector A has a magnitude and B has a magnitude. Imagine a parallogram. Hold the bottom two points in place and rotate the top two points such that they line up with the bottom two points in the vertical axis. Now you have the same magnitudes just at different angles. And the area is not the actual magnitude's products. So you are saying that by shifting the top two points, you are actually changing the area inside the geometric object? It seems counter-intuitive. If you can explain how it does change the area, then I will believe it.

Thanks.
I don't know how to "explain" this to you since you don't seem to want to accept the evidence of your eyes. Instead of rotating so that the top two points "line up with the bottom two points in the vertical axis", rotate so that the lie along the same horizontal line. Do you not see that the area is decreasing toward 0? Yes, of course, changing that angle changes the area.

I don't see any way you can "understand" things like the cross product if you do not know basic geometry.
 
  • #19
By shifting the top two points the magnitudes do not change if you image they are metal rods rotating around the two bottom pivot points. Then you have a rectangle and the magnitudes are exactly the same. So in that case is the area the same as before it was rotated?
Tx
 
  • #20
So you would have maximum volume at 90 degree angles from what you say.
 
  • #21
And of course you could also move the right triangle from the left side and place it on the right, then you would end up with a rectangle whose voluse is width times sin theta time the hypotenuse of the aforementied right triangles. I of course see it that way. And the flattening of the four sides is also very clear as well. Thx
 
  • #22
rockyshephear said:
By shifting the top two points the magnitudes do not change if you image they are metal rods rotating around the two bottom pivot points. Then you have a rectangle and the magnitudes are exactly the same. So in that case is the area the same as before it was rotated?
Tx

No it is not. The argument is by contradiction, as pointed out in HallsofIvy's reply: let your vectors rotate so that the "top points" drop to coincide with the horizontal vector, giving an area of 0, clearly not the non-zero area as they had when a rectangle. The argument that all rotations of the upright vectors contain the same area is therefore not sound.
 
  • #23
Yes, I get that as I stated in the second from the last post. thx
 
  • #24
I just watched an MIT math professor write...
"The length of the cross product = the area of the parallogram with sides A nad B"
So he's say that for example
meters=meters^2
And someone replies and says "Well, he just means the number values are the same!"
I would have to pose the question
Are units not important in many math equations??
One of my profs always used to take off points when no units were used in a math equation.
Thx
 
  • #25
rockyshephear said:
I just watched an MIT math professor …

Are units not important in many math equations??
One of my profs always used to take off points when no units were used in a math equation.
Thx

hmm … fair question :smile: …

well, they're certainly important in physics equations …

but why would a math prof be bothering with units? :confused:

In maths of vector spaces, you usually choose a basis (eg i j k), and then units don't come into it. :wink:

But to return to physics …

angular momentum is position "cross" velocity,

and it's in units of metres squared per second. :smile:​
 
  • #26
Thank Tiny-Tim. Well, when a math teacher is discussing area you would have units specified, for example. :)
But surely physics more so.

So is a cross product vector just another way of stating the answer to a normal physics question
"What is the angular momentum of such and such given the following conditions?"
And the vector in words would be stated
(whatever the magnitude is) metres squared per second?
ie
Vector C is 38 meters per second squared
AND the answer to the physics question just happens to be pointed in 3D space along the axis of rotation?
So you could almost imagine the vectors or whatever they mean rotation around the cross product vector as if it's really an axis of rotation? Or is it merely symbolic with no correlation to a physical rotation around that vector?
I'm going to understand this Cross product thing in real world terms if it kills me. lol
 
  • #27
Oh I should have added...

(whatever the magnitude is) metres squared per second along an infinite amount of vectors all parallel to each other angled in a specific way to the 3D coordinate system.
 
  • #28
rockyshephear said:
… the answer to the physics question just happens to be pointed in 3D space along the axis of rotation?
So you could almost imagine the vectors or whatever they mean rotation around the cross product vector as if it's really an axis of rotation?

You can think of angular momentum as pointy and thin, in which case it's one-dimensional,

or you can think of as circular , in which case it's two-dimensional.

It's the same question as for a rotation …

is a rotation about a line or in a plane? …

you can define it either way. :smile:
 
  • #29
Take for example a gyro.
Once spinning, is it resistance to positional change of the axis that keeps the top upright if started upright? And is that resistance because of the rotating mass around the axis? Can this be defined with vectors as well?
 

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